+ // convergence transformation
+ if (need_trafo) {
+
+ // sort (|x|<->position) to determine indices
+ std::multimap<ex,int> sortmap;
+ int size = 0;
+ for (int i=0; i<x.nops(); ++i) {
+ if (!x[i].is_zero()) {
+ sortmap.insert(std::pair<ex,int>(abs(x[i]), i));
+ ++size;
+ }
+ }
+ // include upper limit (scale)
+ sortmap.insert(std::pair<ex,int>(abs(y), x.nops()));
+
+ // generate missing dummy-symbols
+ int i = 1;
+ gsyms.clear();
+ gsyms.push_back(symbol("GSYMS_ERROR"));
+ ex lastentry;
+ for (std::multimap<ex,int>::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
+ if (it != sortmap.begin()) {
+ if (it->second < x.nops()) {
+ if (x[it->second] == lastentry) {
+ gsyms.push_back(gsyms.back());
+ continue;
+ }
+ } else {
+ if (y == lastentry) {
+ gsyms.push_back(gsyms.back());
+ continue;
+ }
+ }
+ }
+ std::ostringstream os;
+ os << "a" << i;
+ gsyms.push_back(symbol(os.str()));
+ ++i;
+ if (it->second < x.nops()) {
+ lastentry = x[it->second];
+ } else {
+ lastentry = y;
+ }
+ }
+
+ // fill position data according to sorted indices and prepare substitution list
+ Gparameter a(x.nops());
+ lst subslst;
+ int pos = 1;
+ int scale;
+ for (std::multimap<ex,int>::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
+ if (it->second < x.nops()) {
+ if (s[it->second] > 0) {
+ a[it->second] = pos;
+ } else {
+ a[it->second] = -pos;
+ }
+ subslst.append(gsyms[pos] == x[it->second]);
+ } else {
+ scale = pos;
+ subslst.append(gsyms[pos] == y);
+ }
+ ++pos;
+ }
+
+ // do transformation
+ Gparameter pendint;
+ ex result = G_transform(pendint, a, scale);
+ // replace dummy symbols with their values
+ result = result.eval().expand();
+ result = result.subs(subslst).evalf();
+
+ return result;
+ }
+
+ // do acceleration transformation (hoelder convolution [BBB])
+ if (need_hoelder) {
+
+ ex result;
+ const int size = x.nops();
+ lst newx;
+ for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
+ newx.append(*it / y);
+ }
+
+ for (int r=0; r<=size; ++r) {
+ ex buffer = pow(-1, r);
+ ex p = 2;
+ bool adjustp;
+ do {
+ adjustp = false;
+ for (lst::const_iterator it = newx.begin(); it != newx.end(); ++it) {
+ if (*it == 1/p) {
+ p += (3-p)/2;
+ adjustp = true;
+ continue;
+ }
+ }
+ } while (adjustp);
+ ex q = p / (p-1);
+ lst qlstx;
+ lst qlsts;
+ for (int j=r; j>=1; --j) {
+ qlstx.append(1-newx.op(j-1));
+ if (newx.op(j-1).info(info_flags::real) && newx.op(j-1) > 1 && newx.op(j-1) <= 2) {
+ qlsts.append( s.op(j-1));
+ } else {
+ qlsts.append( -s.op(j-1));
+ }
+ }
+ if (qlstx.nops() > 0) {
+ buffer *= G_numeric(qlstx, qlsts, 1/q);
+ }
+ lst plstx;
+ lst plsts;
+ for (int j=r+1; j<=size; ++j) {
+ plstx.append(newx.op(j-1));
+ plsts.append(s.op(j-1));
+ }
+ if (plstx.nops() > 0) {
+ buffer *= G_numeric(plstx, plsts, 1/p);
+ }
+ result += buffer;
+ }
+ return result;
+ }
+
+ // do summation
+ lst newx;
+ lst m;
+ int mcount = 1;
+ ex sign = 1;
+ ex factor = y;
+ for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
+ if ((*it).is_zero()) {
+ ++mcount;
+ } else {
+ newx.append(factor / (*it));
+ factor = *it;
+ m.append(mcount);
+ mcount = 1;
+ sign = -sign;
+ }
+ }
+
+ return sign * numeric(mLi_do_summation(m, newx));
+}
+
+
+ex mLi_numeric(const lst& m, const lst& x)
+{
+ // let G_numeric do the transformation
+ lst newx;
+ lst s;
+ ex factor = 1;
+ for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
+ for (int i = 1; i < *itm; ++i) {
+ newx.append(0);
+ s.append(1);
+ }
+ newx.append(factor / *itx);
+ factor /= *itx;
+ s.append(1);
+ }
+ return pow(-1, m.nops()) * G_numeric(newx, s, _ex1);
+}
+
+
+} // end of anonymous namespace
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// Generalized multiple polylogarithm G(x, y) and G(x, s, y)
+//
+// GiNaC function
+//
+//////////////////////////////////////////////////////////////////////
+
+
+static ex G2_evalf(const ex& x_, const ex& y)
+{
+ if (!y.info(info_flags::positive)) {
+ return G(x_, y).hold();
+ }
+ lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
+ if (x.nops() == 0) {
+ return _ex1;
+ }
+ if (x.op(0) == y) {
+ return G(x_, y).hold();
+ }
+ lst s;
+ bool all_zero = true;
+ for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
+ if (!(*it).info(info_flags::numeric)) {
+ return G(x_, y).hold();
+ }
+ if (*it != _ex0) {
+ all_zero = false;
+ }
+ s.append(+1);
+ }
+ if (all_zero) {
+ return pow(log(y), x.nops()) / factorial(x.nops());
+ }
+ return G_numeric(x, s, y);
+}
+
+
+static ex G2_eval(const ex& x_, const ex& y)
+{
+ //TODO eval to MZV or H or S or Lin
+
+ if (!y.info(info_flags::positive)) {
+ return G(x_, y).hold();
+ }
+ lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
+ if (x.nops() == 0) {
+ return _ex1;
+ }
+ if (x.op(0) == y) {
+ return G(x_, y).hold();
+ }
+ lst s;
+ bool all_zero = true;
+ bool crational = true;
+ for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
+ if (!(*it).info(info_flags::numeric)) {
+ return G(x_, y).hold();
+ }
+ if (!(*it).info(info_flags::crational)) {
+ crational = false;
+ }
+ if (*it != _ex0) {
+ all_zero = false;
+ }
+ s.append(+1);
+ }
+ if (all_zero) {
+ return pow(log(y), x.nops()) / factorial(x.nops());
+ }
+ if (!y.info(info_flags::crational)) {
+ crational = false;
+ }
+ if (crational) {
+ return G(x_, y).hold();
+ }
+ return G_numeric(x, s, y);
+}
+
+
+unsigned G2_SERIAL::serial = function::register_new(function_options("G", 2).
+ evalf_func(G2_evalf).
+ eval_func(G2_eval).
+ do_not_evalf_params().
+ overloaded(2));
+//TODO
+// derivative_func(G2_deriv).
+// print_func<print_latex>(G2_print_latex).
+
+
+static ex G3_evalf(const ex& x_, const ex& s_, const ex& y)
+{
+ if (!y.info(info_flags::positive)) {
+ return G(x_, s_, y).hold();
+ }
+ lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
+ lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
+ if (x.nops() != s.nops()) {
+ return G(x_, s_, y).hold();
+ }
+ if (x.nops() == 0) {
+ return _ex1;
+ }
+ if (x.op(0) == y) {
+ return G(x_, s_, y).hold();
+ }
+ lst sn;
+ bool all_zero = true;
+ for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
+ if (!(*itx).info(info_flags::numeric)) {
+ return G(x_, y).hold();
+ }
+ if (!(*its).info(info_flags::real)) {
+ return G(x_, y).hold();
+ }
+ if (*itx != _ex0) {
+ all_zero = false;
+ }
+ if (*its >= 0) {
+ sn.append(+1);
+ } else {
+ sn.append(-1);
+ }
+ }
+ if (all_zero) {
+ return pow(log(y), x.nops()) / factorial(x.nops());
+ }
+ return G_numeric(x, sn, y);
+}
+
+
+static ex G3_eval(const ex& x_, const ex& s_, const ex& y)
+{
+ //TODO eval to MZV or H or S or Lin
+
+ if (!y.info(info_flags::positive)) {
+ return G(x_, s_, y).hold();
+ }
+ lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
+ lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
+ if (x.nops() != s.nops()) {
+ return G(x_, s_, y).hold();
+ }
+ if (x.nops() == 0) {
+ return _ex1;
+ }
+ if (x.op(0) == y) {
+ return G(x_, s_, y).hold();
+ }
+ lst sn;
+ bool all_zero = true;
+ bool crational = true;
+ for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
+ if (!(*itx).info(info_flags::numeric)) {
+ return G(x_, s_, y).hold();
+ }
+ if (!(*its).info(info_flags::real)) {
+ return G(x_, s_, y).hold();
+ }
+ if (!(*itx).info(info_flags::crational)) {
+ crational = false;
+ }
+ if (*itx != _ex0) {
+ all_zero = false;
+ }
+ if (*its >= 0) {
+ sn.append(+1);
+ } else {
+ sn.append(-1);
+ }
+ }
+ if (all_zero) {
+ return pow(log(y), x.nops()) / factorial(x.nops());
+ }
+ if (!y.info(info_flags::crational)) {
+ crational = false;
+ }
+ if (crational) {
+ return G(x_, s_, y).hold();
+ }
+ return G_numeric(x, sn, y);
+}
+
+
+unsigned G3_SERIAL::serial = function::register_new(function_options("G", 3).
+ evalf_func(G3_evalf).
+ eval_func(G3_eval).
+ do_not_evalf_params().
+ overloaded(2));
+//TODO
+// derivative_func(G3_deriv).
+// print_func<print_latex>(G3_print_latex).
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// Classical polylogarithm and multiple polylogarithm Li(m,x)
+//
+// GiNaC function
+//
+//////////////////////////////////////////////////////////////////////
+
+
+static ex Li_evalf(const ex& m_, const ex& x_)
+{
+ // classical polylogs
+ if (m_.info(info_flags::posint)) {
+ if (x_.info(info_flags::numeric)) {
+ return Lin_numeric(ex_to<numeric>(m_).to_int(), ex_to<numeric>(x_));
+ } else {
+ // try to numerically evaluate second argument
+ ex x_val = x_.evalf();
+ if (x_val.info(info_flags::numeric)) {
+ return Lin_numeric(ex_to<numeric>(m_).to_int(), ex_to<numeric>(x_val));
+ }
+ }
+ }
+ // multiple polylogs
+ if (is_a<lst>(m_) && is_a<lst>(x_)) {
+
+ const lst& m = ex_to<lst>(m_);
+ const lst& x = ex_to<lst>(x_);
+ if (m.nops() != x.nops()) {
+ return Li(m_,x_).hold();
+ }
+ if (x.nops() == 0) {
+ return _ex1;
+ }
+ if ((m.op(0) == _ex1) && (x.op(0) == _ex1)) {
+ return Li(m_,x_).hold();
+ }
+
+ for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
+ if (!(*itm).info(info_flags::posint)) {
+ return Li(m_, x_).hold();
+ }
+ if (!(*itx).info(info_flags::numeric)) {
+ return Li(m_, x_).hold();
+ }
+ if (*itx == _ex0) {
+ return _ex0;
+ }
+ }
+
+ return mLi_numeric(m, x);
+ }
+
+ return Li(m_,x_).hold();
+}
+
+
+static ex Li_eval(const ex& m_, const ex& x_)
+{
+ if (is_a<lst>(m_)) {
+ if (is_a<lst>(x_)) {
+ // multiple polylogs
+ const lst& m = ex_to<lst>(m_);
+ const lst& x = ex_to<lst>(x_);
+ if (m.nops() != x.nops()) {
+ return Li(m_,x_).hold();
+ }
+ if (x.nops() == 0) {
+ return _ex1;
+ }
+ bool is_H = true;
+ bool is_zeta = true;
+ bool do_evalf = true;
+ bool crational = true;
+ for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
+ if (!(*itm).info(info_flags::posint)) {
+ return Li(m_,x_).hold();
+ }
+ if ((*itx != _ex1) && (*itx != _ex_1)) {
+ if (itx != x.begin()) {
+ is_H = false;
+ }
+ is_zeta = false;
+ }
+ if (*itx == _ex0) {
+ return _ex0;
+ }
+ if (!(*itx).info(info_flags::numeric)) {
+ do_evalf = false;
+ }
+ if (!(*itx).info(info_flags::crational)) {
+ crational = false;
+ }
+ }
+ if (is_zeta) {
+ return zeta(m_,x_);
+ }
+ if (is_H) {
+ ex prefactor;
+ lst newm = convert_parameter_Li_to_H(m, x, prefactor);
+ return prefactor * H(newm, x[0]);
+ }
+ if (do_evalf && !crational) {
+ return mLi_numeric(m,x);
+ }
+ }
+ return Li(m_, x_).hold();
+ } else if (is_a<lst>(x_)) {
+ return Li(m_, x_).hold();
+ }
+
+ // classical polylogs
+ if (x_ == _ex0) {
+ return _ex0;
+ }
+ if (x_ == _ex1) {
+ return zeta(m_);
+ }
+ if (x_ == _ex_1) {
+ return (pow(2,1-m_)-1) * zeta(m_);
+ }
+ if (m_ == _ex1) {
+ return -log(1-x_);
+ }
+ if (m_ == _ex2) {
+ if (x_.is_equal(I)) {
+ return power(Pi,_ex2)/_ex_48 + Catalan*I;
+ }
+ if (x_.is_equal(-I)) {
+ return power(Pi,_ex2)/_ex_48 - Catalan*I;
+ }
+ }
+ if (m_.info(info_flags::posint) && x_.info(info_flags::numeric) && !x_.info(info_flags::crational)) {
+ return Lin_numeric(ex_to<numeric>(m_).to_int(), ex_to<numeric>(x_));
+ }
+
+ return Li(m_, x_).hold();
+}
+
+
+static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
+{
+ if (is_a<lst>(m) || is_a<lst>(x)) {
+ // multiple polylog
+ epvector seq;
+ seq.push_back(expair(Li(m, x), 0));
+ return pseries(rel, seq);
+ }
+
+ // classical polylog
+ const ex x_pt = x.subs(rel, subs_options::no_pattern);
+ if (m.info(info_flags::numeric) && x_pt.info(info_flags::numeric)) {
+ // First special case: x==0 (derivatives have poles)
+ if (x_pt.is_zero()) {
+ const symbol s;
+ ex ser;
+ // manually construct the primitive expansion
+ for (int i=1; i<order; ++i)
+ ser += pow(s,i) / pow(numeric(i), m);
+ // substitute the argument's series expansion
+ ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
+ // maybe that was terminating, so add a proper order term
+ epvector nseq;
+ nseq.push_back(expair(Order(_ex1), order));
+ ser += pseries(rel, nseq);
+ // reexpanding it will collapse the series again
+ return ser.series(rel, order);
+ }
+ // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
+ throw std::runtime_error("Li_series: don't know how to do the series expansion at this point!");
+ }
+ // all other cases should be safe, by now:
+ throw do_taylor(); // caught by function::series()
+}
+
+
+static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param)
+{
+ GINAC_ASSERT(deriv_param < 2);
+ if (deriv_param == 0) {
+ return _ex0;
+ }
+ if (m_.nops() > 1) {
+ throw std::runtime_error("don't know how to derivate multiple polylogarithm!");
+ }
+ ex m;
+ if (is_a<lst>(m_)) {
+ m = m_.op(0);
+ } else {
+ m = m_;
+ }
+ ex x;
+ if (is_a<lst>(x_)) {
+ x = x_.op(0);
+ } else {
+ x = x_;
+ }
+ if (m > 0) {
+ return Li(m-1, x) / x;
+ } else {
+ return 1/(1-x);
+ }
+}
+
+
+static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
+{
+ lst m;
+ if (is_a<lst>(m_)) {
+ m = ex_to<lst>(m_);
+ } else {
+ m = lst(m_);
+ }
+ lst x;
+ if (is_a<lst>(x_)) {
+ x = ex_to<lst>(x_);
+ } else {
+ x = lst(x_);
+ }
+ c.s << "\\mbox{Li}_{";
+ lst::const_iterator itm = m.begin();
+ (*itm).print(c);
+ itm++;
+ for (; itm != m.end(); itm++) {
+ c.s << ",";
+ (*itm).print(c);
+ }
+ c.s << "}(";
+ lst::const_iterator itx = x.begin();
+ (*itx).print(c);
+ itx++;
+ for (; itx != x.end(); itx++) {
+ c.s << ",";
+ (*itx).print(c);
+ }
+ c.s << ")";
+}
+
+
+REGISTER_FUNCTION(Li,
+ evalf_func(Li_evalf).
+ eval_func(Li_eval).
+ series_func(Li_series).
+ derivative_func(Li_deriv).
+ print_func<print_latex>(Li_print_latex).
+ do_not_evalf_params());
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// Nielsen's generalized polylogarithm S(n,p,x)
+//
+// helper functions
+//
+//////////////////////////////////////////////////////////////////////
+
+
+// anonymous namespace for helper functions
+namespace {
+
+
+// lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
+// see fill_Yn()
+std::vector<std::vector<cln::cl_N> > Yn;
+int ynsize = 0; // number of Yn[]
+int ynlength = 100; // initial length of all Yn[i]
+
+
+// This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
+// The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
+// representing S_{n,p}(x).
+// The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
+// equivalent Z-sum.
+// The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
+// representing S_{n,p}(x).
+// The calculation of Y_n uses the values from Y_{n-1}.
+void fill_Yn(int n, const cln::float_format_t& prec)
+{
+ const int initsize = ynlength;
+ //const int initsize = initsize_Yn;
+ cln::cl_N one = cln::cl_float(1, prec);
+
+ if (n) {
+ std::vector<cln::cl_N> buf(initsize);
+ std::vector<cln::cl_N>::iterator it = buf.begin();
+ std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
+ *it = (*itprev) / cln::cl_N(n+1) * one;
+ it++;
+ itprev++;
+ // sums with an index smaller than the depth are zero and need not to be calculated.
+ // calculation starts with depth, which is n+2)
+ for (int i=n+2; i<=initsize+n; i++) {
+ *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
+ it++;
+ itprev++;
+ }
+ Yn.push_back(buf);
+ } else {
+ std::vector<cln::cl_N> buf(initsize);
+ std::vector<cln::cl_N>::iterator it = buf.begin();
+ *it = 1 * one;
+ it++;
+ for (int i=2; i<=initsize; i++) {
+ *it = *(it-1) + 1 / cln::cl_N(i) * one;
+ it++;
+ }
+ Yn.push_back(buf);
+ }
+ ynsize++;
+}
+
+
+// make Yn longer ...
+void make_Yn_longer(int newsize, const cln::float_format_t& prec)
+{
+
+ cln::cl_N one = cln::cl_float(1, prec);
+
+ Yn[0].resize(newsize);
+ std::vector<cln::cl_N>::iterator it = Yn[0].begin();
+ it += ynlength;
+ for (int i=ynlength+1; i<=newsize; i++) {
+ *it = *(it-1) + 1 / cln::cl_N(i) * one;
+ it++;
+ }
+
+ for (int n=1; n<ynsize; n++) {
+ Yn[n].resize(newsize);
+ std::vector<cln::cl_N>::iterator it = Yn[n].begin();
+ std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
+ it += ynlength;
+ itprev += ynlength;
+ for (int i=ynlength+n+1; i<=newsize+n; i++) {
+ *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
+ it++;
+ itprev++;
+ }
+ }
+
+ ynlength = newsize;
+}
+
+
+// helper function for S(n,p,x)
+// [Kol] (7.2)
+cln::cl_N C(int n, int p)
+{
+ cln::cl_N result;
+
+ for (int k=0; k<p; k++) {
+ for (int j=0; j<=(n+k-1)/2; j++) {
+ if (k == 0) {
+ if (n & 1) {
+ if (j & 1) {
+ result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
+ }
+ else {
+ result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
+ }
+ }
+ }
+ else {
+ if (k & 1) {
+ if (j & 1) {
+ result = result + cln::factorial(n+k-1)
+ * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
+ / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
+ }
+ else {
+ result = result - cln::factorial(n+k-1)
+ * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
+ / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
+ }
+ }
+ else {
+ if (j & 1) {
+ result = result - cln::factorial(n+k-1) * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
+ / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
+ }
+ else {
+ result = result + cln::factorial(n+k-1)
+ * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
+ / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
+ }
+ }
+ }
+ }
+ }
+ int np = n+p;
+ if ((np-1) & 1) {
+ if (((np)/2+n) & 1) {
+ result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
+ }
+ else {
+ result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
+ }
+ }
+
+ return result;
+}
+
+
+// helper function for S(n,p,x)
+// [Kol] remark to (9.1)
+cln::cl_N a_k(int k)
+{
+ cln::cl_N result;
+
+ if (k == 0) {
+ return 1;
+ }
+
+ result = result;
+ for (int m=2; m<=k; m++) {
+ result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
+ }
+
+ return -result / k;
+}
+
+
+// helper function for S(n,p,x)
+// [Kol] remark to (9.1)
+cln::cl_N b_k(int k)
+{
+ cln::cl_N result;
+
+ if (k == 0) {
+ return 1;
+ }
+
+ result = result;
+ for (int m=2; m<=k; m++) {
+ result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
+ }
+
+ return result / k;
+}
+
+
+// helper function for S(n,p,x)
+cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
+{
+ if (p==1) {
+ return Li_projection(n+1, x, prec);
+ }
+
+ // check if precalculated values are sufficient
+ if (p > ynsize+1) {
+ for (int i=ynsize; i<p-1; i++) {
+ fill_Yn(i, prec);
+ }
+ }
+
+ // should be done otherwise
+ cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
+ cln::cl_N xf = x * one;
+ //cln::cl_N xf = x * cln::cl_float(1, prec);
+
+ cln::cl_N res;
+ cln::cl_N resbuf;
+ cln::cl_N factor = cln::expt(xf, p);
+ int i = p;
+ do {
+ resbuf = res;
+ if (i-p >= ynlength) {
+ // make Yn longer
+ make_Yn_longer(ynlength*2, prec);
+ }
+ res = res + factor / cln::expt(cln::cl_I(i),n+1) * Yn[p-2][i-p]; // should we check it? or rely on magic number? ...
+ //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
+ factor = factor * xf;
+ i++;
+ } while (res != resbuf);
+
+ return res;
+}
+
+
+// helper function for S(n,p,x)
+cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
+{
+ // [Kol] (5.3)
+ if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
+
+ cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
+ * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
+
+ for (int s=0; s<n; s++) {
+ cln::cl_N res2;
+ for (int r=0; r<p; r++) {
+ res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
+ * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
+ }
+ result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
+ }
+
+ return result;
+ }
+
+ return S_do_sum(n, p, x, prec);
+}
+
+
+// helper function for S(n,p,x)
+numeric S_num(int n, int p, const numeric& x)
+{
+ if (x == 1) {
+ if (n == 1) {
+ // [Kol] (2.22) with (2.21)
+ return cln::zeta(p+1);
+ }
+
+ if (p == 1) {
+ // [Kol] (2.22)
+ return cln::zeta(n+1);
+ }
+
+ // [Kol] (9.1)
+ cln::cl_N result;
+ for (int nu=0; nu<n; nu++) {
+ for (int rho=0; rho<=p; rho++) {
+ result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
+ * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
+ }
+ }
+ result = result * cln::expt(cln::cl_I(-1),n+p-1);
+
+ return result;
+ }
+ else if (x == -1) {
+ // [Kol] (2.22)
+ if (p == 1) {
+ return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
+ }
+// throw std::runtime_error("don't know how to evaluate this function!");
+ }
+
+ // what is the desired float format?
+ // first guess: default format
+ cln::float_format_t prec = cln::default_float_format;
+ const cln::cl_N value = x.to_cl_N();
+ // second guess: the argument's format
+ if (!x.real().is_rational())
+ prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
+ else if (!x.imag().is_rational())
+ prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
+
+ // [Kol] (5.3)
+ if ((cln::realpart(value) < -0.5) || (n == 0)) {
+
+ cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
+ * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
+
+ for (int s=0; s<n; s++) {
+ cln::cl_N res2;
+ for (int r=0; r<p; r++) {
+ res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
+ * S_num(p-r,n-s,1-value).to_cl_N() / cln::factorial(r);
+ }
+ result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
+ }
+
+ return result;
+
+ }
+ // [Kol] (5.12)
+ if (cln::abs(value) > 1) {
+
+ cln::cl_N result;
+
+ for (int s=0; s<p; s++) {
+ for (int r=0; r<=s; r++) {
+ result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
+ / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
+ * S_num(n+s-r,p-s,cln::recip(value)).to_cl_N();
+ }
+ }
+ result = result * cln::expt(cln::cl_I(-1),n);
+
+ cln::cl_N res2;
+ for (int r=0; r<n; r++) {
+ res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
+ }
+ res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
+
+ result = result + cln::expt(cln::cl_I(-1),p) * res2;
+
+ return result;
+ }
+ else {
+ return S_projection(n, p, value, prec);
+ }
+}
+
+
+} // end of anonymous namespace
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// Nielsen's generalized polylogarithm S(n,p,x)
+//
+// GiNaC function
+//
+//////////////////////////////////////////////////////////////////////
+
+
+static ex S_evalf(const ex& n, const ex& p, const ex& x)
+{
+ if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
+ if (is_a<numeric>(x)) {
+ return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
+ } else {
+ ex x_val = x.evalf();
+ if (is_a<numeric>(x_val)) {
+ return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x_val));
+ }
+ }
+ }
+ return S(n, p, x).hold();
+}
+
+
+static ex S_eval(const ex& n, const ex& p, const ex& x)
+{
+ if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
+ if (x == 0) {
+ return _ex0;
+ }
+ if (x == 1) {
+ lst m(n+1);
+ for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
+ m.append(1);
+ }
+ return zeta(m);
+ }
+ if (p == 1) {
+ return Li(n+1, x);
+ }
+ if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
+ return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
+ }
+ }
+ if (n.is_zero()) {
+ // [Kol] (5.3)
+ return pow(-log(1-x), p) / factorial(p);
+ }
+ return S(n, p, x).hold();
+}
+
+
+static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
+{
+ if (p == _ex1) {
+ return Li(n+1, x).series(rel, order, options);
+ }
+
+ const ex x_pt = x.subs(rel, subs_options::no_pattern);
+ if (n.info(info_flags::posint) && p.info(info_flags::posint) && x_pt.info(info_flags::numeric)) {
+ // First special case: x==0 (derivatives have poles)
+ if (x_pt.is_zero()) {
+ const symbol s;
+ ex ser;
+ // manually construct the primitive expansion
+ // subsum = Euler-Zagier-Sum is needed
+ // dirty hack (slow ...) calculation of subsum:
+ std::vector<ex> presubsum, subsum;
+ subsum.push_back(0);
+ for (int i=1; i<order-1; ++i) {
+ subsum.push_back(subsum[i-1] + numeric(1, i));
+ }
+ for (int depth=2; depth<p; ++depth) {
+ presubsum = subsum;
+ for (int i=1; i<order-1; ++i) {
+ subsum[i] = subsum[i-1] + numeric(1, i) * presubsum[i-1];
+ }
+ }
+
+ for (int i=1; i<order; ++i) {
+ ser += pow(s,i) / pow(numeric(i), n+1) * subsum[i-1];
+ }
+ // substitute the argument's series expansion
+ ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
+ // maybe that was terminating, so add a proper order term
+ epvector nseq;
+ nseq.push_back(expair(Order(_ex1), order));
+ ser += pseries(rel, nseq);
+ // reexpanding it will collapse the series again
+ return ser.series(rel, order);
+ }
+ // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
+ throw std::runtime_error("S_series: don't know how to do the series expansion at this point!");
+ }
+ // all other cases should be safe, by now:
+ throw do_taylor(); // caught by function::series()
+}
+
+
+static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
+{
+ GINAC_ASSERT(deriv_param < 3);
+ if (deriv_param < 2) {
+ return _ex0;
+ }
+ if (n > 0) {
+ return S(n-1, p, x) / x;
+ } else {
+ return S(n, p-1, x) / (1-x);
+ }
+}
+
+
+static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
+{
+ c.s << "\\mbox{S}_{";
+ n.print(c);
+ c.s << ",";
+ p.print(c);
+ c.s << "}(";
+ x.print(c);
+ c.s << ")";
+}
+
+
+REGISTER_FUNCTION(S,
+ evalf_func(S_evalf).
+ eval_func(S_eval).
+ series_func(S_series).
+ derivative_func(S_deriv).
+ print_func<print_latex>(S_print_latex).
+ do_not_evalf_params());
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// Harmonic polylogarithm H(m,x)
+//
+// helper functions
+//
+//////////////////////////////////////////////////////////////////////
+
+
+// anonymous namespace for helper functions
+namespace {
+
+
+// regulates the pole (used by 1/x-transformation)
+symbol H_polesign("IMSIGN");
+
+
+// convert parameters from H to Li representation
+// parameters are expected to be in expanded form, i.e. only 0, 1 and -1
+// returns true if some parameters are negative
+bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
+{
+ // expand parameter list
+ lst mexp;
+ for (lst::const_iterator it = l.begin(); it != l.end(); it++) {
+ if (*it > 1) {
+ for (ex count=*it-1; count > 0; count--) {
+ mexp.append(0);
+ }
+ mexp.append(1);
+ } else if (*it < -1) {
+ for (ex count=*it+1; count < 0; count++) {
+ mexp.append(0);
+ }
+ mexp.append(-1);
+ } else {
+ mexp.append(*it);
+ }
+ }
+
+ ex signum = 1;
+ pf = 1;
+ bool has_negative_parameters = false;
+ ex acc = 1;
+ for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) {
+ if (*it == 0) {
+ acc++;
+ continue;
+ }
+ if (*it > 0) {
+ m.append((*it+acc-1) * signum);
+ } else {
+ m.append((*it-acc+1) * signum);
+ }
+ acc = 1;
+ signum = *it;
+ pf *= *it;
+ if (pf < 0) {
+ has_negative_parameters = true;
+ }
+ }
+ if (has_negative_parameters) {
+ for (int i=0; i<m.nops(); i++) {
+ if (m.op(i) < 0) {
+ m.let_op(i) = -m.op(i);
+ s.append(-1);
+ } else {
+ s.append(1);
+ }
+ }
+ }
+
+ return has_negative_parameters;
+}
+
+
+// recursivly transforms H to corresponding multiple polylogarithms
+struct map_trafo_H_convert_to_Li : public map_function
+{
+ ex operator()(const ex& e)
+ {
+ if (is_a<add>(e) || is_a<mul>(e)) {
+ return e.map(*this);
+ }
+ if (is_a<function>(e)) {
+ std::string name = ex_to<function>(e).get_name();
+ if (name == "H") {
+ lst parameter;
+ if (is_a<lst>(e.op(0))) {
+ parameter = ex_to<lst>(e.op(0));
+ } else {
+ parameter = lst(e.op(0));
+ }
+ ex arg = e.op(1);
+
+ lst m;
+ lst s;
+ ex pf;
+ if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
+ s.let_op(0) = s.op(0) * arg;
+ return pf * Li(m, s).hold();
+ } else {
+ for (int i=0; i<m.nops(); i++) {
+ s.append(1);
+ }
+ s.let_op(0) = s.op(0) * arg;
+ return Li(m, s).hold();
+ }
+ }
+ }
+ return e;
+ }
+};
+
+
+// recursivly transforms H to corresponding zetas
+struct map_trafo_H_convert_to_zeta : public map_function
+{
+ ex operator()(const ex& e)
+ {
+ if (is_a<add>(e) || is_a<mul>(e)) {
+ return e.map(*this);
+ }
+ if (is_a<function>(e)) {
+ std::string name = ex_to<function>(e).get_name();
+ if (name == "H") {
+ lst parameter;
+ if (is_a<lst>(e.op(0))) {
+ parameter = ex_to<lst>(e.op(0));
+ } else {
+ parameter = lst(e.op(0));
+ }
+
+ lst m;
+ lst s;
+ ex pf;
+ if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
+ return pf * zeta(m, s);
+ } else {
+ return zeta(m);
+ }
+ }
+ }
+ return e;
+ }
+};
+
+
+// remove trailing zeros from H-parameters
+struct map_trafo_H_reduce_trailing_zeros : public map_function
+{
+ ex operator()(const ex& e)
+ {
+ if (is_a<add>(e) || is_a<mul>(e)) {
+ return e.map(*this);
+ }
+ if (is_a<function>(e)) {
+ std::string name = ex_to<function>(e).get_name();
+ if (name == "H") {
+ lst parameter;
+ if (is_a<lst>(e.op(0))) {
+ parameter = ex_to<lst>(e.op(0));
+ } else {
+ parameter = lst(e.op(0));
+ }
+ ex arg = e.op(1);
+ if (parameter.op(parameter.nops()-1) == 0) {
+
+ //
+ if (parameter.nops() == 1) {
+ return log(arg);
+ }
+
+ //
+ lst::const_iterator it = parameter.begin();
+ while ((it != parameter.end()) && (*it == 0)) {
+ it++;
+ }
+ if (it == parameter.end()) {
+ return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
+ }
+
+ //
+ parameter.remove_last();
+ int lastentry = parameter.nops();
+ while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
+ lastentry--;
+ }
+
+ //
+ ex result = log(arg) * H(parameter,arg).hold();
+ ex acc = 0;
+ for (ex i=0; i<lastentry; i++) {
+ if (parameter[i] > 0) {
+ parameter[i]++;
+ result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
+ parameter[i]--;
+ acc = 0;
+ } else if (parameter[i] < 0) {
+ parameter[i]--;
+ result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
+ parameter[i]++;
+ acc = 0;
+ } else {
+ acc++;
+ }
+ }
+
+ if (lastentry < parameter.nops()) {
+ result = result / (parameter.nops()-lastentry+1);
+ return result.map(*this);
+ } else {
+ return result;
+ }
+ }
+ }
+ }
+ return e;
+ }
+};
+
+
+// returns an expression with zeta functions corresponding to the parameter list for H
+ex convert_H_to_zeta(const lst& m)
+{
+ symbol xtemp("xtemp");
+ map_trafo_H_reduce_trailing_zeros filter;
+ map_trafo_H_convert_to_zeta filter2;
+ return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
+}
+
+
+// convert signs form Li to H representation
+lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
+{
+ lst res;
+ lst::const_iterator itm = m.begin();
+ lst::const_iterator itx = ++x.begin();
+ int signum = 1;
+ pf = _ex1;
+ res.append(*itm);
+ itm++;
+ while (itx != x.end()) {
+ signum *= (*itx > 0) ? 1 : -1;
+ pf *= signum;
+ res.append((*itm) * signum);
+ itm++;
+ itx++;
+ }
+ return res;
+}
+
+
+// multiplies an one-dimensional H with another H
+// [ReV] (18)
+ex trafo_H_mult(const ex& h1, const ex& h2)
+{
+ ex res;
+ ex hshort;
+ lst hlong;
+ ex h1nops = h1.op(0).nops();
+ ex h2nops = h2.op(0).nops();
+ if (h1nops > 1) {
+ hshort = h2.op(0).op(0);
+ hlong = ex_to<lst>(h1.op(0));
+ } else {
+ hshort = h1.op(0).op(0);
+ if (h2nops > 1) {
+ hlong = ex_to<lst>(h2.op(0));
+ } else {
+ hlong = h2.op(0).op(0);
+ }
+ }
+ for (int i=0; i<=hlong.nops(); i++) {
+ lst newparameter;
+ int j=0;
+ for (; j<i; j++) {
+ newparameter.append(hlong[j]);
+ }
+ newparameter.append(hshort);
+ for (; j<hlong.nops(); j++) {
+ newparameter.append(hlong[j]);
+ }
+ res += H(newparameter, h1.op(1)).hold();
+ }
+ return res;
+}
+
+
+// applies trafo_H_mult recursively on expressions
+struct map_trafo_H_mult : public map_function
+{
+ ex operator()(const ex& e)
+ {
+ if (is_a<add>(e)) {
+ return e.map(*this);
+ }
+
+ if (is_a<mul>(e)) {
+
+ ex result = 1;
+ ex firstH;
+ lst Hlst;
+ for (int pos=0; pos<e.nops(); pos++) {
+ if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
+ std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
+ if (name == "H") {
+ for (ex i=0; i<e.op(pos).op(1); i++) {
+ Hlst.append(e.op(pos).op(0));
+ }
+ continue;
+ }
+ } else if (is_a<function>(e.op(pos))) {
+ std::string name = ex_to<function>(e.op(pos)).get_name();
+ if (name == "H") {
+ if (e.op(pos).op(0).nops() > 1) {
+ firstH = e.op(pos);
+ } else {
+ Hlst.append(e.op(pos));
+ }
+ continue;
+ }
+ }
+ result *= e.op(pos);
+ }
+ if (firstH == 0) {
+ if (Hlst.nops() > 0) {
+ firstH = Hlst[Hlst.nops()-1];
+ Hlst.remove_last();
+ } else {
+ return e;
+ }
+ }
+
+ if (Hlst.nops() > 0) {
+ ex buffer = trafo_H_mult(firstH, Hlst.op(0));
+ result *= buffer;
+ for (int i=1; i<Hlst.nops(); i++) {
+ result *= Hlst.op(i);
+ }
+ result = result.expand();
+ map_trafo_H_mult recursion;
+ return recursion(result);
+ } else {
+ return e;
+ }
+
+ }
+ return e;
+ }
+};
+
+
+// do integration [ReV] (55)
+// put parameter 0 in front of existing parameters
+ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
+{
+ ex h;
+ std::string name;
+ if (is_a<function>(e)) {
+ name = ex_to<function>(e).get_name();
+ }
+ if (name == "H") {
+ h = e;
+ } else {
+ for (int i=0; i<e.nops(); i++) {
+ if (is_a<function>(e.op(i))) {
+ std::string name = ex_to<function>(e.op(i)).get_name();
+ if (name == "H") {
+ h = e.op(i);
+ }
+ }
+ }
+ }
+ if (h != 0) {
+ lst newparameter = ex_to<lst>(h.op(0));
+ newparameter.prepend(0);
+ ex addzeta = convert_H_to_zeta(newparameter);
+ return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
+ } else {
+ return e * (-H(lst(0),1/arg).hold());
+ }
+}
+
+
+// do integration [ReV] (49)
+// put parameter 1 in front of existing parameters
+ex trafo_H_prepend_one(const ex& e, const ex& arg)
+{
+ ex h;
+ std::string name;
+ if (is_a<function>(e)) {
+ name = ex_to<function>(e).get_name();
+ }
+ if (name == "H") {
+ h = e;
+ } else {
+ for (int i=0; i<e.nops(); i++) {
+ if (is_a<function>(e.op(i))) {
+ std::string name = ex_to<function>(e.op(i)).get_name();
+ if (name == "H") {
+ h = e.op(i);
+ }
+ }
+ }
+ }
+ if (h != 0) {
+ lst newparameter = ex_to<lst>(h.op(0));
+ newparameter.prepend(1);
+ return e.subs(h == H(newparameter, h.op(1)).hold());
+ } else {
+ return e * H(lst(1),1-arg).hold();
+ }
+}
+
+
+// do integration [ReV] (55)
+// put parameter -1 in front of existing parameters
+ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
+{
+ ex h;
+ std::string name;
+ if (is_a<function>(e)) {
+ name = ex_to<function>(e).get_name();
+ }
+ if (name == "H") {
+ h = e;
+ } else {
+ for (int i=0; i<e.nops(); i++) {
+ if (is_a<function>(e.op(i))) {
+ std::string name = ex_to<function>(e.op(i)).get_name();
+ if (name == "H") {
+ h = e.op(i);
+ }
+ }
+ }
+ }
+ if (h != 0) {
+ lst newparameter = ex_to<lst>(h.op(0));
+ newparameter.prepend(-1);
+ ex addzeta = convert_H_to_zeta(newparameter);
+ return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
+ } else {
+ ex addzeta = convert_H_to_zeta(lst(-1));
+ return (e * (addzeta - H(lst(-1),1/arg).hold())).expand();
+ }
+}
+
+
+// do integration [ReV] (55)
+// put parameter -1 in front of existing parameters
+ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
+{
+ ex h;
+ std::string name;
+ if (is_a<function>(e)) {
+ name = ex_to<function>(e).get_name();
+ }
+ if (name == "H") {
+ h = e;
+ } else {
+ for (int i=0; i<e.nops(); i++) {
+ if (is_a<function>(e.op(i))) {
+ std::string name = ex_to<function>(e.op(i)).get_name();
+ if (name == "H") {
+ h = e.op(i);
+ }
+ }
+ }
+ }
+ if (h != 0) {
+ lst newparameter = ex_to<lst>(h.op(0));
+ newparameter.prepend(-1);
+ return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
+ } else {
+ return (e * H(lst(-1),(1-arg)/(1+arg)).hold()).expand();
+ }
+}
+
+
+// do integration [ReV] (55)
+// put parameter 1 in front of existing parameters
+ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
+{
+ ex h;
+ std::string name;
+ if (is_a<function>(e)) {
+ name = ex_to<function>(e).get_name();
+ }
+ if (name == "H") {
+ h = e;
+ } else {
+ for (int i=0; i<e.nops(); i++) {
+ if (is_a<function>(e.op(i))) {
+ std::string name = ex_to<function>(e.op(i)).get_name();
+ if (name == "H") {
+ h = e.op(i);
+ }
+ }
+ }
+ }
+ if (h != 0) {
+ lst newparameter = ex_to<lst>(h.op(0));
+ newparameter.prepend(1);
+ return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
+ } else {
+ return (e * H(lst(1),(1-arg)/(1+arg)).hold()).expand();
+ }
+}
+
+
+// do x -> 1-x transformation
+struct map_trafo_H_1mx : public map_function
+{
+ ex operator()(const ex& e)
+ {
+ if (is_a<add>(e) || is_a<mul>(e)) {
+ return e.map(*this);
+ }
+
+ if (is_a<function>(e)) {
+ std::string name = ex_to<function>(e).get_name();
+ if (name == "H") {
+
+ lst parameter = ex_to<lst>(e.op(0));
+ ex arg = e.op(1);
+
+ // special cases if all parameters are either 0, 1 or -1
+ bool allthesame = true;
+ if (parameter.op(0) == 0) {
+ for (int i=1; i<parameter.nops(); i++) {
+ if (parameter.op(i) != 0) {
+ allthesame = false;
+ break;
+ }
+ }
+ if (allthesame) {
+ lst newparameter;
+ for (int i=parameter.nops(); i>0; i--) {
+ newparameter.append(0);
+ }
+ return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
+ }
+ } else if (parameter.op(0) == -1) {
+ throw std::runtime_error("map_trafo_H_1mx: cannot handle weights equal -1!");
+ } else {
+ for (int i=1; i<parameter.nops(); i++) {
+ if (parameter.op(i) != 1) {
+ allthesame = false;
+ break;
+ }
+ }
+ if (allthesame) {
+ lst newparameter;
+ for (int i=parameter.nops(); i>0; i--) {
+ newparameter.append(1);
+ }
+ return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
+ }
+ }
+
+ lst newparameter = parameter;
+ newparameter.remove_first();
+
+ if (parameter.op(0) == 0) {
+
+ // leading zero
+ ex res = convert_H_to_zeta(parameter);
+ //ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
+ map_trafo_H_1mx recursion;
+ ex buffer = recursion(H(newparameter, arg).hold());
+ if (is_a<add>(buffer)) {
+ for (int i=0; i<buffer.nops(); i++) {
+ res -= trafo_H_prepend_one(buffer.op(i), arg);
+ }
+ } else {
+ res -= trafo_H_prepend_one(buffer, arg);
+ }
+ return res;
+
+ } else {
+
+ // leading one
+ map_trafo_H_1mx recursion;
+ map_trafo_H_mult unify;
+ ex res;
+ int firstzero = 0;
+ while (parameter.op(firstzero) == 1) {
+ firstzero++;
+ }
+ for (int i=firstzero-1; i<parameter.nops()-1; i++) {
+ lst newparameter;
+ int j=0;
+ for (; j<=i; j++) {
+ newparameter.append(parameter[j+1]);
+ }
+ newparameter.append(1);
+ for (; j<parameter.nops()-1; j++) {
+ newparameter.append(parameter[j+1]);
+ }
+ res -= H(newparameter, arg).hold();
+ }
+ return (unify((-H(lst(0), 1-arg).hold() * recursion(H(newparameter, arg).hold())).expand()) +
+ recursion(res)) / firstzero;
+
+ }
+
+ }
+ }
+ return e;
+ }
+};
+
+
+// do x -> 1/x transformation
+struct map_trafo_H_1overx : public map_function
+{
+ ex operator()(const ex& e)
+ {
+ if (is_a<add>(e) || is_a<mul>(e)) {
+ return e.map(*this);
+ }
+
+ if (is_a<function>(e)) {
+ std::string name = ex_to<function>(e).get_name();
+ if (name == "H") {
+
+ lst parameter = ex_to<lst>(e.op(0));
+ ex arg = e.op(1);
+
+ // special cases if all parameters are either 0, 1 or -1
+ bool allthesame = true;
+ if (parameter.op(0) == 0) {
+ for (int i=1; i<parameter.nops(); i++) {
+ if (parameter.op(i) != 0) {
+ allthesame = false;
+ break;
+ }
+ }
+ if (allthesame) {
+ return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
+ }
+ } else if (parameter.op(0) == -1) {
+ for (int i=1; i<parameter.nops(); i++) {
+ if (parameter.op(i) != -1) {
+ allthesame = false;
+ break;
+ }
+ }
+ if (allthesame) {
+ map_trafo_H_mult unify;
+ return unify((pow(H(lst(-1),1/arg).hold() - H(lst(0),1/arg).hold(), parameter.nops())
+ / factorial(parameter.nops())).expand());
+ }
+ } else {
+ for (int i=1; i<parameter.nops(); i++) {
+ if (parameter.op(i) != 1) {
+ allthesame = false;
+ break;
+ }
+ }
+ if (allthesame) {
+ map_trafo_H_mult unify;
+ return unify((pow(H(lst(1),1/arg).hold() + H(lst(0),1/arg).hold() + H_polesign, parameter.nops())
+ / factorial(parameter.nops())).expand());
+ }
+ }
+
+ lst newparameter = parameter;
+ newparameter.remove_first();
+
+ if (parameter.op(0) == 0) {
+
+ // leading zero
+ ex res = convert_H_to_zeta(parameter);
+ map_trafo_H_1overx recursion;
+ ex buffer = recursion(H(newparameter, arg).hold());
+ if (is_a<add>(buffer)) {
+ for (int i=0; i<buffer.nops(); i++) {
+ res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
+ }
+ } else {
+ res += trafo_H_1tx_prepend_zero(buffer, arg);
+ }
+ return res;
+
+ } else if (parameter.op(0) == -1) {
+
+ // leading negative one
+ ex res = convert_H_to_zeta(parameter);
+ map_trafo_H_1overx recursion;
+ ex buffer = recursion(H(newparameter, arg).hold());
+ if (is_a<add>(buffer)) {
+ for (int i=0; i<buffer.nops(); i++) {
+ res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
+ }
+ } else {
+ res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
+ }
+ return res;
+
+ } else {
+
+ // leading one
+ map_trafo_H_1overx recursion;
+ map_trafo_H_mult unify;
+ ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
+ int firstzero = 0;
+ while (parameter.op(firstzero) == 1) {
+ firstzero++;
+ }
+ for (int i=firstzero-1; i<parameter.nops()-1; i++) {
+ lst newparameter;
+ int j=0;
+ for (; j<=i; j++) {
+ newparameter.append(parameter[j+1]);
+ }
+ newparameter.append(1);
+ for (; j<parameter.nops()-1; j++) {
+ newparameter.append(parameter[j+1]);
+ }
+ res -= H(newparameter, arg).hold();
+ }
+ res = recursion(res).expand() / firstzero;
+ return unify(res);
+
+ }
+
+ }
+ }
+ return e;
+ }
+};
+
+
+// do x -> (1-x)/(1+x) transformation
+struct map_trafo_H_1mxt1px : public map_function
+{
+ ex operator()(const ex& e)
+ {
+ if (is_a<add>(e) || is_a<mul>(e)) {
+ return e.map(*this);
+ }
+
+ if (is_a<function>(e)) {
+ std::string name = ex_to<function>(e).get_name();
+ if (name == "H") {
+
+ lst parameter = ex_to<lst>(e.op(0));
+ ex arg = e.op(1);
+
+ // special cases if all parameters are either 0, 1 or -1
+ bool allthesame = true;
+ if (parameter.op(0) == 0) {
+ for (int i=1; i<parameter.nops(); i++) {
+ if (parameter.op(i) != 0) {
+ allthesame = false;
+ break;
+ }
+ }
+ if (allthesame) {
+ map_trafo_H_mult unify;
+ return unify((pow(-H(lst(1),(1-arg)/(1+arg)).hold() - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
+ / factorial(parameter.nops())).expand());
+ }
+ } else if (parameter.op(0) == -1) {
+ for (int i=1; i<parameter.nops(); i++) {
+ if (parameter.op(i) != -1) {
+ allthesame = false;
+ break;
+ }
+ }
+ if (allthesame) {
+ map_trafo_H_mult unify;
+ return unify((pow(log(2) - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
+ / factorial(parameter.nops())).expand());
+ }
+ } else {
+ for (int i=1; i<parameter.nops(); i++) {
+ if (parameter.op(i) != 1) {
+ allthesame = false;
+ break;
+ }
+ }
+ if (allthesame) {
+ map_trafo_H_mult unify;
+ return unify((pow(-log(2) - H(lst(0),(1-arg)/(1+arg)).hold() + H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
+ / factorial(parameter.nops())).expand());
+ }
+ }
+
+ lst newparameter = parameter;
+ newparameter.remove_first();
+
+ if (parameter.op(0) == 0) {
+
+ // leading zero
+ ex res = convert_H_to_zeta(parameter);
+ map_trafo_H_1mxt1px recursion;
+ ex buffer = recursion(H(newparameter, arg).hold());
+ if (is_a<add>(buffer)) {
+ for (int i=0; i<buffer.nops(); i++) {
+ res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
+ }
+ } else {
+ res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
+ }
+ return res;
+
+ } else if (parameter.op(0) == -1) {
+
+ // leading negative one
+ ex res = convert_H_to_zeta(parameter);
+ map_trafo_H_1mxt1px recursion;
+ ex buffer = recursion(H(newparameter, arg).hold());
+ if (is_a<add>(buffer)) {
+ for (int i=0; i<buffer.nops(); i++) {
+ res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
+ }
+ } else {
+ res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
+ }
+ return res;
+
+ } else {
+
+ // leading one
+ map_trafo_H_1mxt1px recursion;
+ map_trafo_H_mult unify;
+ ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
+ int firstzero = 0;
+ while (parameter.op(firstzero) == 1) {
+ firstzero++;
+ }
+ for (int i=firstzero-1; i<parameter.nops()-1; i++) {
+ lst newparameter;
+ int j=0;
+ for (; j<=i; j++) {
+ newparameter.append(parameter[j+1]);
+ }
+ newparameter.append(1);
+ for (; j<parameter.nops()-1; j++) {
+ newparameter.append(parameter[j+1]);
+ }
+ res -= H(newparameter, arg).hold();
+ }
+ res = recursion(res).expand() / firstzero;
+ return unify(res);
+
+ }
+
+ }
+ }
+ return e;
+ }
+};
+
+
+// do the actual summation.
+cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
+{
+ const int j = m.size();
+
+ std::vector<cln::cl_N> t(j);
+
+ cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
+ cln::cl_N factor = cln::expt(x, j) * one;
+ cln::cl_N t0buf;
+ int q = 0;
+ do {
+ t0buf = t[0];
+ q++;
+ t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
+ for (int k=j-2; k>=1; k--) {
+ t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
+ }
+ t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
+ factor = factor * x;
+ } while (t[0] != t0buf);
+
+ return t[0];
+}
+
+
+} // end of anonymous namespace
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// Harmonic polylogarithm H(m,x)
+//
+// GiNaC function
+//
+//////////////////////////////////////////////////////////////////////
+
+
+static ex H_evalf(const ex& x1, const ex& x2)
+{
+ if (is_a<lst>(x1)) {
+
+ cln::cl_N x;
+ if (is_a<numeric>(x2)) {
+ x = ex_to<numeric>(x2).to_cl_N();
+ } else {
+ ex x2_val = x2.evalf();
+ if (is_a<numeric>(x2_val)) {
+ x = ex_to<numeric>(x2_val).to_cl_N();
+ }
+ }
+
+ for (int i=0; i<x1.nops(); i++) {
+ if (!x1.op(i).info(info_flags::integer)) {
+ return H(x1, x2).hold();
+ }
+ }
+ if (x1.nops() < 1) {
+ return H(x1, x2).hold();
+ }
+
+ const lst& morg = ex_to<lst>(x1);
+ // remove trailing zeros ...
+ if (*(--morg.end()) == 0) {
+ symbol xtemp("xtemp");
+ map_trafo_H_reduce_trailing_zeros filter;
+ return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
+ }
+ // ... and expand parameter notation
+ bool has_minus_one = false;
+ lst m;
+ for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) {
+ if (*it > 1) {
+ for (ex count=*it-1; count > 0; count--) {
+ m.append(0);
+ }
+ m.append(1);
+ } else if (*it <= -1) {
+ for (ex count=*it+1; count < 0; count++) {
+ m.append(0);
+ }
+ m.append(-1);
+ has_minus_one = true;
+ } else {
+ m.append(*it);
+ }
+ }
+
+ // do summation
+ if (cln::abs(x) < 0.95) {
+ lst m_lst;
+ lst s_lst;
+ ex pf;
+ if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
+ // negative parameters -> s_lst is filled
+ std::vector<int> m_int;
+ std::vector<cln::cl_N> x_cln;
+ for (lst::const_iterator it_int = m_lst.begin(), it_cln = s_lst.begin();
+ it_int != m_lst.end(); it_int++, it_cln++) {
+ m_int.push_back(ex_to<numeric>(*it_int).to_int());
+ x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
+ }
+ x_cln.front() = x_cln.front() * x;
+ return pf * numeric(multipleLi_do_sum(m_int, x_cln));
+ } else {
+ // only positive parameters
+ //TODO
+ if (m_lst.nops() == 1) {
+ return Li(m_lst.op(0), x2).evalf();
+ }
+ std::vector<int> m_int;
+ for (lst::const_iterator it = m_lst.begin(); it != m_lst.end(); it++) {
+ m_int.push_back(ex_to<numeric>(*it).to_int());
+ }
+ return numeric(H_do_sum(m_int, x));
+ }
+ }
+
+ symbol xtemp("xtemp");
+ ex res = 1;
+
+ // ensure that the realpart of the argument is positive
+ if (cln::realpart(x) < 0) {
+ x = -x;
+ for (int i=0; i<m.nops(); i++) {
+ if (m.op(i) != 0) {
+ m.let_op(i) = -m.op(i);
+ res *= -1;
+ }
+ }
+ }
+
+ // x -> 1/x
+ if (cln::abs(x) >= 2.0) {
+ map_trafo_H_1overx trafo;
+ res *= trafo(H(m, xtemp));
+ if (cln::imagpart(x) <= 0) {
+ res = res.subs(H_polesign == -I*Pi);
+ } else {
+ res = res.subs(H_polesign == I*Pi);
+ }
+ return res.subs(xtemp == numeric(x)).evalf();
+ }
+
+ // check transformations for 0.95 <= |x| < 2.0
+
+ // |(1-x)/(1+x)| < 0.9 -> circular area with center=9,53+0i and radius=9.47
+ if (cln::abs(x-9.53) <= 9.47) {
+ // x -> (1-x)/(1+x)
+ map_trafo_H_1mxt1px trafo;
+ res *= trafo(H(m, xtemp));
+ } else {
+ // x -> 1-x
+ if (has_minus_one) {
+ map_trafo_H_convert_to_Li filter;
+ return filter(H(m, numeric(x)).hold()).evalf();
+ }
+ map_trafo_H_1mx trafo;
+ res *= trafo(H(m, xtemp));
+ }
+
+ return res.subs(xtemp == numeric(x)).evalf();
+ }
+
+ return H(x1,x2).hold();
+}
+
+
+static ex H_eval(const ex& m_, const ex& x)
+{
+ lst m;
+ if (is_a<lst>(m_)) {
+ m = ex_to<lst>(m_);
+ } else {
+ m = lst(m_);
+ }
+ if (m.nops() == 0) {
+ return _ex1;
+ }
+ ex pos1;
+ ex pos2;
+ ex n;
+ ex p;
+ int step = 0;
+ if (*m.begin() > _ex1) {
+ step++;
+ pos1 = _ex0;
+ pos2 = _ex1;
+ n = *m.begin()-1;
+ p = _ex1;
+ } else if (*m.begin() < _ex_1) {
+ step++;
+ pos1 = _ex0;
+ pos2 = _ex_1;
+ n = -*m.begin()-1;
+ p = _ex1;
+ } else if (*m.begin() == _ex0) {
+ pos1 = _ex0;
+ n = _ex1;
+ } else {
+ pos1 = *m.begin();
+ p = _ex1;
+ }
+ for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) {
+ if ((*it).info(info_flags::integer)) {
+ if (step == 0) {
+ if (*it > _ex1) {
+ if (pos1 == _ex0) {
+ step = 1;
+ pos2 = _ex1;
+ n += *it-1;
+ p = _ex1;
+ } else {
+ step = 2;
+ }
+ } else if (*it < _ex_1) {
+ if (pos1 == _ex0) {
+ step = 1;
+ pos2 = _ex_1;
+ n += -*it-1;
+ p = _ex1;
+ } else {
+ step = 2;
+ }
+ } else {
+ if (*it != pos1) {
+ step = 1;
+ pos2 = *it;
+ }
+ if (*it == _ex0) {
+ n++;
+ } else {
+ p++;
+ }
+ }
+ } else if (step == 1) {
+ if (*it != pos2) {
+ step = 2;
+ } else {
+ if (*it == _ex0) {
+ n++;
+ } else {
+ p++;
+ }
+ }
+ }
+ } else {
+ // if some m_i is not an integer
+ return H(m_, x).hold();
+ }
+ }
+ if ((x == _ex1) && (*(--m.end()) != _ex0)) {
+ return convert_H_to_zeta(m);
+ }
+ if (step == 0) {
+ if (pos1 == _ex0) {
+ // all zero
+ if (x == _ex0) {
+ return H(m_, x).hold();
+ }
+ return pow(log(x), m.nops()) / factorial(m.nops());
+ } else {
+ // all (minus) one
+ return pow(-pos1*log(1-pos1*x), m.nops()) / factorial(m.nops());
+ }
+ } else if ((step == 1) && (pos1 == _ex0)){
+ // convertible to S
+ if (pos2 == _ex1) {
+ return S(n, p, x);
+ } else {
+ return pow(-1, p) * S(n, p, -x);
+ }
+ }
+ if (x == _ex0) {
+ return _ex0;
+ }
+ if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
+ return H(m_, x).evalf();
+ }
+ return H(m_, x).hold();
+}
+
+
+static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
+{
+ epvector seq;
+ seq.push_back(expair(H(m, x), 0));
+ return pseries(rel, seq);
+}
+
+
+static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param)
+{
+ GINAC_ASSERT(deriv_param < 2);
+ if (deriv_param == 0) {
+ return _ex0;
+ }
+ lst m;
+ if (is_a<lst>(m_)) {
+ m = ex_to<lst>(m_);
+ } else {
+ m = lst(m_);
+ }
+ ex mb = *m.begin();
+ if (mb > _ex1) {
+ m[0]--;
+ return H(m, x) / x;
+ }
+ if (mb < _ex_1) {
+ m[0]++;
+ return H(m, x) / x;
+ }
+ m.remove_first();
+ if (mb == _ex1) {
+ return 1/(1-x) * H(m, x);
+ } else if (mb == _ex_1) {
+ return 1/(1+x) * H(m, x);
+ } else {
+ return H(m, x) / x;
+ }
+}
+
+
+static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
+{
+ lst m;
+ if (is_a<lst>(m_)) {
+ m = ex_to<lst>(m_);
+ } else {
+ m = lst(m_);
+ }
+ c.s << "\\mbox{H}_{";
+ lst::const_iterator itm = m.begin();
+ (*itm).print(c);
+ itm++;
+ for (; itm != m.end(); itm++) {
+ c.s << ",";
+ (*itm).print(c);
+ }
+ c.s << "}(";
+ x.print(c);
+ c.s << ")";
+}
+
+
+REGISTER_FUNCTION(H,
+ evalf_func(H_evalf).
+ eval_func(H_eval).
+ series_func(H_series).
+ derivative_func(H_deriv).
+ print_func<print_latex>(H_print_latex).
+ do_not_evalf_params());
+
+
+// takes a parameter list for H and returns an expression with corresponding multiple polylogarithms
+ex convert_H_to_Li(const ex& m, const ex& x)
+{
+ map_trafo_H_reduce_trailing_zeros filter;
+ map_trafo_H_convert_to_Li filter2;
+ if (is_a<lst>(m)) {
+ return filter2(filter(H(m, x).hold()));
+ } else {
+ return filter2(filter(H(lst(m), x).hold()));
+ }
+}
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// Multiple zeta values zeta(x) and zeta(x,s)
+//
+// helper functions
+//
+//////////////////////////////////////////////////////////////////////
+
+
+// anonymous namespace for helper functions
+namespace {
+
+
+// parameters and data for [Cra] algorithm
+const cln::cl_N lambda = cln::cl_N("319/320");
+int L1;
+int L2;
+std::vector<std::vector<cln::cl_N> > f_kj;
+std::vector<cln::cl_N> crB;
+std::vector<std::vector<cln::cl_N> > crG;
+std::vector<cln::cl_N> crX;
+
+
+void halfcyclic_convolute(const std::vector<cln::cl_N>& a, const std::vector<cln::cl_N>& b, std::vector<cln::cl_N>& c)
+{
+ const int size = a.size();
+ for (int n=0; n<size; n++) {
+ c[n] = 0;
+ for (int m=0; m<=n; m++) {
+ c[n] = c[n] + a[m]*b[n-m];
+ }
+ }