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Is there a short form for summation of following series?

$$\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(-1)^kn!\alpha^{2n}((2y-1)^{2k+1}+1)}{2^{2n+1}(2n)!k!(n-k)!(2k+1)}$$ $$\sum\limits_{n=0}^\infty\dfrac{\alpha^{2n+1}(\cos^{-1}(2y-1)-\pi)}{2^{4n+3}n!(n+1)!}$$ $$\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(k!)^2\alpha^{2n+1}(2y-1)(1-(2y-1)^2)^{k+\frac{1}{2}}}{2^{4n-2k+3}n!(n+1)!(2k+1)!}$$

Even sum of any combination of above terms could help. This is result of some integral. I guess they should contain Bessel and Struve functions. In the case of $y=0.5$ it seems the sum of above terms to be

$$ -\frac{\pi}{2}\left[I_3(\frac{\alpha}{2}) +\frac{3}{\frac{\alpha}{2}}I_2(\frac{\alpha}{2})- I_{1}(\frac{\alpha}{2})-\frac{1}{8}\left[L_{-3}(\frac{\alpha}{2}) - L_{-1}(\frac{\alpha}{2}) - L_{1}(\frac{\alpha}{2})+L_{3}(\frac{\alpha}{2}) -\frac{2\alpha^{-2}}{\pi} -\frac{8}{3\pi}+\frac{2\alpha^2}{15\pi}\right]\right] $$

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    $\begingroup$ The second one equals $$\frac{\cos^{-1}(2y-1)-\pi}{2}\,I_1(\alpha/2)$$ where $I_1$ is a Bessel function. $\endgroup$ Aug 23, 2014 at 22:12
  • $\begingroup$ Many thanks for letting me know $\endgroup$
    – Roy
    Aug 23, 2014 at 22:26

2 Answers 2

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For the first series, I was able to wittle it down to a sum of a Struve function and an unevaluated series of hypergeometric functions:

$$\begin{align} S{(\alpha,y)} &=\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(-1)^kn!\alpha^{2n}((2y-1)^{2k+1}+1)}{2^{2n+1}(2n)!k!(n-k)!(2k+1)}\\ &=\sum\limits_{n=0}^\infty \frac{\alpha^{2n}}{2^{2n+1}(2n)!} \sum\limits_{k=0}^n (-1)^k \frac{n!}{k!(n-k)!} \dfrac{((2y-1)^{2k+1}+1)}{(2k+1)}\\ &=\sum\limits_{n=0}^\infty \frac{\alpha^{2n}}{2^{2n+1}(2n)!} \left[\sum\limits_{k=0}^n \dfrac{(-1)^k\binom{n}{k}}{(2k+1)} + \sum\limits_{k=0}^n \dfrac{(-1)^k\binom{n}{k}}{(2k+1)} (2y-1)^{2k+1}\right]\\ &=\sum\limits_{n=0}^\infty \frac{\alpha^{2n}}{2^{2n+1}(2n)!} \left[\frac{(2n)!!}{(2n+1)!!} + (2y-1)\,{_2F_1}{\left(\frac12,-n;\frac32;(2y-1)^2\right)}\right]\\ &=\sum\limits_{n=0}^\infty \frac{\alpha^{2n}}{2^{2n+1}(2n)!}\frac{(2n)!!}{(2n+1)!!} + (2y-1)\sum\limits_{n=0}^\infty \frac{\alpha^{2n}}{2^{2n+1}(2n)!} {_2F_1}{\left(\frac12,-n;\frac32;(2y-1)^2\right)}\\ &=\frac{\pi}{4}\operatorname{L}_{-1}{\left(\frac{\alpha}{2}\right)} + (2y-1)\sum\limits_{n=0}^\infty \frac{\alpha^{2n}}{2^{2n+1}(2n)!} {_2F_1}{\left(\frac12,-n;\frac32;(2y-1)^2\right)}. \end{align}$$

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    $\begingroup$ Note that the hypergeometric functions included in the final answer are essentially Jacobi polynomials with $(\alpha,\beta)=(1/2,-n-1)$: $$P_n^{(1/2,-n-1)}(z)=\dfrac{(3/2)_n}{n!} {_2F_1}{\left(-n,\frac12,\frac32;\frac{1}{2}(1-z)\right)}.$$ So that last series can possibly be refined further... $\endgroup$ Aug 24, 2014 at 2:16
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For $\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(-1)^kn!\alpha^{2n}(2y-1)^{2k+1}}{2^{2n+1}(2n)!k!(n-k)!(2k+1)}$ ,

$\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(-1)^kn!\alpha^{2n}(2y-1)^{2k+1}}{2^{2n+1}(2n)!k!(n-k)!(2k+1)}$

$=\sum\limits_{k=0}^\infty\sum\limits_{n=k}^\infty\dfrac{(-1)^kn!\alpha^{2n}(2y-1)^{2k+1}}{2^{2n+1}(2n)!k!(n-k)!(2k+1)}$

$=\sum\limits_{k=0}^\infty\sum\limits_{n=0}^\infty\dfrac{(-1)^k(n+k)!\alpha^{2n+2k}(2y-1)^{2k+1}}{2^{2n+2k+1}(2n+2k)!k!n!(2k+1)}$

$=\sum\limits_{k=0}^\infty\sum\limits_{n=0}^\infty\dfrac{(-1)^k\alpha^{2n+2k}(2y-1)^{2k+1}\sqrt\pi}{2^{4n+4k+1}\Gamma\left(n+k+\dfrac{1}{2}\right)n!k!\left(k+\dfrac{1}{2}\right)}$ (according to https://en.wikipedia.org/wiki/Multiplication_theorem#Gamma_function-Legendre_function)

For $\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(k!)^2\alpha^{2n+1}(2y-1)(1-(2y-1)^2)^{k+\frac{1}{2}}}{2^{4n-2k+3}n!(n+1)!(2k+1)!}$ ,

$\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(k!)^2\alpha^{2n+1}(2y-1)(1-(2y-1)^2)^{k+\frac{1}{2}}}{2^{4n-2k+3}n!(n+1)!(2k+1)!}$

$=\sum\limits_{k=0}^\infty\sum\limits_{n=k}^\infty\dfrac{(k!)^2\alpha^{2n+1}(2y-1)(1-(2y-1)^2)^{k+\frac{1}{2}}}{2^{4n-2k+3}n!(n+1)!(2k+1)!}$

$=\sum\limits_{k=0}^\infty\sum\limits_{n=0}^\infty\dfrac{(k!)^2\alpha^{2n+2k+1}(2y-1)(1-(2y-1)^2)^{k+\frac{1}{2}}}{2^{4n+2k+3}(n+k)!(n+k+1)!(2k+1)!}$

$=\sum\limits_{k=0}^\infty\sum\limits_{n=0}^\infty\dfrac{(k!)^2\alpha^{2n+2k+1}(2y-1)(1-(2y-1)^2)^{k+\frac{1}{2}}\sqrt\pi}{16^{n+k+1}(n+k)!(n+k+1)!\Gamma(k+1)\Gamma\left(k+\dfrac{3}{2}\right)}$ (according to https://en.wikipedia.org/wiki/Multiplication_theorem#Gamma_function-Legendre_function)

which both relate to Kampé de Fériet function function

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