pFqToStandardFunctions - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

SumTools[DefiniteSum]

 pFqToStandardFunctions
 compute closed forms of definite sums using hypergeometric functions

 Calling Sequence pFqToStandardFunctions(f, k=m..n)

Parameters

 f - expression; specified summand k - name m, n - expressions or integers

Description

 • 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).

Examples

 > $\mathrm{with}\left({\mathrm{SumTools}}_{\mathrm{DefiniteSum}}\right):$
 > $F≔\frac{{2}^{2k}\mathrm{GAMMA}\left(k-n\right)\mathrm{GAMMA}\left(k+n\right){z}^{k}}{{\mathrm{Pi}}^{\frac{1}{2}}\mathrm{GAMMA}\left(2k+1\right)}$
 ${F}{≔}\frac{{{2}}^{{2}{}{k}}{}{\mathrm{\Gamma }}{}\left({k}{-}{n}\right){}{\mathrm{\Gamma }}{}\left({k}{+}{n}\right){}{{z}}^{{k}}}{\sqrt{{\mathrm{\pi }}}{}{\mathrm{\Gamma }}{}\left({2}{}{k}{+}{1}\right)}$ (1)
 > $\mathrm{pFqToStandardFunctions}\left(F,k=0..\mathrm{∞}\right)$
 ${\mathrm{FAIL}}$ (2)
 > ${\sum }_{k=0}^{\mathrm{∞}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}F=\mathrm{pFqToStandardFunctions}\left(F,k=0..\mathrm{∞}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\left|z\right|\le 1$
 ${\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{{2}}^{{2}{}{k}}{}{\mathrm{\Gamma }}{}\left({k}{-}{n}\right){}{\mathrm{\Gamma }}{}\left({k}{+}{n}\right){}{{z}}^{{k}}}{\sqrt{{\mathrm{\pi }}}{}{\mathrm{\Gamma }}{}\left({2}{}{k}{+}{1}\right)}{=}{-}\frac{\sqrt{{\mathrm{\pi }}}{}{\mathrm{cos}}{}\left({2}{}{n}{}{\mathrm{arcsin}}{}\left(\sqrt{{z}}\right)\right){}{\mathrm{csc}}{}\left({\mathrm{\pi }}{}{n}\right)}{{n}}$ (3)
 > $\mathrm{_EnvFormal}≔\mathrm{true}:$
 > ${\sum }_{k=0}^{\mathrm{∞}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}F=\mathrm{pFqToStandardFunctions}\left(F,k=0..\mathrm{∞}\right)$
 ${\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{{2}}^{{2}{}{k}}{}{\mathrm{\Gamma }}{}\left({k}{-}{n}\right){}{\mathrm{\Gamma }}{}\left({k}{+}{n}\right){}{{z}}^{{k}}}{\sqrt{{\mathrm{\pi }}}{}{\mathrm{\Gamma }}{}\left({2}{}{k}{+}{1}\right)}{=}{-}\frac{\sqrt{{\mathrm{\pi }}}{}{\mathrm{cos}}{}\left({2}{}{n}{}{\mathrm{arcsin}}{}\left(\sqrt{{z}}\right)\right){}{\mathrm{csc}}{}\left({\mathrm{\pi }}{}{n}\right)}{{n}}$ (4)

References

 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.