- // psi(0,x) -> psi(x)
- if (n.is_zero())
- return psi(x);
- // psi(-1,x) -> log(gamma(x))
- if (n.is_equal(exMINUSONE()))
- return log(gamma(x));
- if (n.info(info_flags::numeric) && n.info(info_flags::posint) &&
- x.info(info_flags::numeric)) {
- numeric nn = ex_to_numeric(n);
- numeric nx = ex_to_numeric(x);
- if (x.is_equal(exONE()))
- return numMINUSONE().power(nn+numONE())*factorial(nn)*zeta(ex(nn+numONE()));
- }
- return psi(n, x).hold();
+ // psi(0,x) -> psi(x)
+ if (n.is_zero())
+ return psi(x);
+ // psi(-1,x) -> log(tgamma(x))
+ if (n.is_equal(_ex_1))
+ return log(tgamma(x));
+ if (n.info(info_flags::numeric) && n.info(info_flags::posint) &&
+ x.info(info_flags::numeric)) {
+ const numeric &nn = ex_to<numeric>(n);
+ const numeric &nx = ex_to<numeric>(x);
+ if (nx.is_integer()) {
+ // integer case
+ if (nx.is_equal(*_num1_p))
+ // use psi(n,1) == (-)^(n+1) * n! * zeta(n+1)
+ return pow(*_num_1_p,nn+(*_num1_p))*factorial(nn)*zeta(ex(nn+(*_num1_p)));
+ if (nx.is_positive()) {
+ // use the recurrence relation
+ // psi(n,m) == psi(n,m+1) - (-)^n * n! / m^(n+1)
+ // to relate psi(n,m) to psi(n,1):
+ // psi(n,m) == psi(n,1) + r
+ // where r == (-)^n * n! * (1^(-n-1) + ... + (m-1)^(-n-1))
+ numeric recur = 0;
+ for (numeric p = 1; p<nx; ++p)
+ recur += pow(p, -nn+(*_num_1_p));
+ recur *= factorial(nn)*pow((*_num_1_p), nn);
+ return recur+psi(n,_ex1);
+ } else {
+ // for non-positive integers there is a pole:
+ throw (pole_error("psi2_eval(): pole",1));
+ }
+ }
+ if (((*_num2_p)*nx).is_integer()) {
+ // half integer case
+ if (nx.is_equal(*_num1_2_p))
+ // use psi(n,1/2) == (-)^(n+1) * n! * (2^(n+1)-1) * zeta(n+1)
+ return pow(*_num_1_p,nn+(*_num1_p))*factorial(nn)*(pow(*_num2_p,nn+(*_num1_p)) + (*_num_1_p))*zeta(ex(nn+(*_num1_p)));
+ if (nx.is_positive()) {
+ const numeric m = nx - (*_num1_2_p);
+ // use the multiplication formula
+ // psi(n,2*m) == (psi(n,m) + psi(n,m+1/2)) / 2^(n+1)
+ // to revert to positive integer case
+ return psi(n,(*_num2_p)*m)*pow((*_num2_p),nn+(*_num1_p))-psi(n,m);
+ } else {
+ // use the recurrence relation
+ // psi(n,-m-1/2) == psi(n,-m-1/2+1) - (-)^n * n! / (-m-1/2)^(n+1)
+ // to relate psi(n,-m-1/2) to psi(n,1/2):
+ // psi(n,-m-1/2) == psi(n,1/2) + r
+ // where r == (-)^(n+1) * n! * ((-1/2)^(-n-1) + ... + (-m-1/2)^(-n-1))
+ numeric recur = 0;
+ for (numeric p = nx; p<0; ++p)
+ recur += pow(p, -nn+(*_num_1_p));
+ recur *= factorial(nn)*pow(*_num_1_p, nn+(*_num_1_p));
+ return recur+psi(n,_ex1_2);
+ }
+ }
+ // psi2_evalf should be called here once it becomes available
+ }
+
+ return psi(n, x).hold();