The pFqToStandardFunctions(f, k=m..n) command computes a closed form of the definite sum of f over the specified range of k by first converting the specified sum into hypergeometric functions. If possible, the output is then converted to standard functions.

If _EnvFormal is assigned true, the command computes the result in the sense of analytic continuation. Otherwise, the command computes the closed form in the domain of convergence (see hypergeom).

$\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{DefiniteSum}\right]\right)\:$

$F≔\frac{\frac{2^{2k}}{{\mathrm{\pi }}^{\frac{1}{2}}}\mathrm{\Gamma }\left(k-n\right)\mathrm{\Gamma }\left(k+n\right)}{\mathrm{\Gamma }\left(2k+1\right)}{z}^k$

$F≔\frac{2^{2k}\mathrm{\Gamma }\left(k-n\right)\mathrm{\Gamma }\left(k+n\right)z^k}{\sqrt{\mathrm{\pi }}\mathrm{\Gamma }\left(2k+1\right)}$

$\mathrm{pFqToStandardFunctions}\left(F\,k=0..\mathrm{\infty }\right)$

$\mathrm{FAIL}$

$\mathrm{Sum}\left(F\,k=0..\mathrm{\infty }\right)=\mathrm{pFqToStandardFunctions}\left(F\,k=0..\mathrm{\infty }\right)\mathbf{assuming}\mathrm{abs}\left(z\right)\le 1$

$\sum _{k=0}^{\mathrm{\infty }}\frac{2^{2k}\mathrm{\Gamma }\left(k-n\right)\mathrm{\Gamma }\left(k+n\right)z^k}{\sqrt{\mathrm{\pi }}\mathrm{\Gamma }\left(2k+1\right)}=-\frac{\sqrt{\mathrm{\pi }}\cos \left(2n\arcsin \left(\sqrt{z}\right)\right)\csc \left(\mathrm{\pi }n\right)}{n}$

$\mathrm{\_EnvFormal}≔\mathrm{true}\:$

$\mathrm{Sum}\left(F\,k=0..\mathrm{\infty }\right)=\mathrm{pFqToStandardFunctions}\left(F\,k=0..\mathrm{\infty }\right)$

$\sum _{k=0}^{\mathrm{\infty }}\frac{2^{2k}\mathrm{\Gamma }\left(k-n\right)\mathrm{\Gamma }\left(k+n\right)z^k}{\sqrt{\mathrm{\pi }}\mathrm{\Gamma }\left(2k+1\right)}=-\frac{\sqrt{\mathrm{\pi }}\cos \left(2n\arcsin \left(\sqrt{z}\right)\right)\csc \left(\mathrm{\pi }n\right)}{n}$

Petkovsek, M.; Wilf, H.; and Zeilberger, D. A=B, Ch. 3. Wellesley, Massachusetts: A K Peters, Ltd., 1996.

Prudnikov, A. P.; Brychkov, Yu.; and Marichev, O. Integrals and Series. Gordon and Breach Science Publishers, 1990. Vol. 3: More Special Functions.

Roach, K. "Hypergeometric Function Representations." Proceedings ISSAC 1996, pp. 301-308. New York: ACM Press, 1996.