pFqToStandardFunctions - Maple Help

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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

 > $\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{DefiniteSum}\right]\right):$
 > $F≔\frac{\frac{{2}^{2k}}{{\mathrm{\pi }}^{\frac{1}{2}}}\mathrm{\Gamma }\left(k-n\right)\mathrm{\Gamma }\left(k+n\right)}{\mathrm{\Gamma }\left(2k+1\right)}{z}^{k}$
 ${F}{≔}\frac{{{2}}^{{2}{}{k}}{}{\mathrm{\Gamma }}{}\left({k}{-}{n}\right){}{\mathrm{\Gamma }}{}\left({k}{+}{n}\right){}{{z}}^{{k}}}{\sqrt{{\mathrm{\pi }}}{}{\mathrm{\Gamma }}{}\left({2}{}{k}{+}{1}\right)}$ (1)
 > $\mathrm{pFqToStandardFunctions}\left(F,k=0..\mathrm{\infty }\right)$
 ${\mathrm{FAIL}}$ (2)
 > $\mathrm{Sum}\left(F,k=0..\mathrm{\infty }\right)=\mathrm{pFqToStandardFunctions}\left(F,k=0..\mathrm{\infty }\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{abs}\left(z\right)\le 1$
 ${\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{{2}}^{{2}{}{k}}{}{\mathrm{\Gamma }}{}\left({k}{-}{n}\right){}{\mathrm{\Gamma }}{}\left({k}{+}{n}\right){}{{z}}^{{k}}}{\sqrt{{\mathrm{\pi }}}{}{\mathrm{\Gamma }}{}\left({2}{}{k}{+}{1}\right)}{=}{-}\frac{\sqrt{{\mathrm{\pi }}}{}{\mathrm{cos}}{}\left({2}{}{n}{}{\mathrm{arcsin}}{}\left(\sqrt{{z}}\right)\right){}{\mathrm{csc}}{}\left({\mathrm{\pi }}{}{n}\right)}{{n}}$ (3)
 > $\mathrm{_EnvFormal}≔\mathrm{true}:$
 > $\mathrm{Sum}\left(F,k=0..\mathrm{\infty }\right)=\mathrm{pFqToStandardFunctions}\left(F,k=0..\mathrm{\infty }\right)$
 ${\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{{2}}^{{2}{}{k}}{}{\mathrm{\Gamma }}{}\left({k}{-}{n}\right){}{\mathrm{\Gamma }}{}\left({k}{+}{n}\right){}{{z}}^{{k}}}{\sqrt{{\mathrm{\pi }}}{}{\mathrm{\Gamma }}{}\left({2}{}{k}{+}{1}\right)}{=}{-}\frac{\sqrt{{\mathrm{\pi }}}{}{\mathrm{cos}}{}\left({2}{}{n}{}{\mathrm{arcsin}}{}\left(\sqrt{{z}}\right)\right){}{\mathrm{csc}}{}\left({\mathrm{\pi }}{}{n}\right)}{{n}}$ (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.

 See Also