Closed form for $\sum_{k=0}^\infty \frac{x^{2 k+1}}{(2k+1) \Gamma (\frac{2k+1}{q})}$

Question

I've been trying to answer another question, and as a part of the solution encountered the following series:

$$S_q(x)=\sum_{k=0}^\infty \frac{x^{2 k+1}}{(2k+1) \Gamma (\frac{2k+1}{q})}$$

Where $q=1,2,3,4,\ldots$.

For $q=1$ and $q=2$ the closed form is trivial, however, for $q \geq 3$ it becomes harder and harder to find.

I would like to know the general form, which should look something like this:

$$S_q(x)= \sum_{n=1}^q F_n(x)$$ Where $F_n(x)$ is a hypergeometric function, possibly multiplied by some factors.

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Expressing:

$$2k+1=qm+n$$

We have:

$$\sum_{m=0}^\infty \frac{x^{qm+n}}{(qm+n) \Gamma (m+\frac{n}{q})}= \frac{x^n}{n \Gamma(\frac{n}{q})} {_1 F_1} \left(1; 1+ \frac{n}{q}; x^q \right)$$

And the series in general should reduce to a finite sum of such terms, but I don't know how to correctly describe all the possible choices of $n$ for a particular $q$.

Answer 1

First, multiply by $q$ to simplify the denominator:

$$qS_q(x)=\sum_{k=0}^\infty\frac{x^{2k+1}}{\Gamma\left[\frac{2k+q+1}q\right]}$$

If all you want is hypergeometric functions, then just look at the ratio between terms. Let $k=qm+n$ and $a_{m+1}/a_m$:

$$\frac{x^{2qm+2n+2q+1}}{\Gamma\left[\frac{2qm+2n+3q+1}q\right]}\div\frac{\Gamma\left[\frac{2qm+2n+q+1}q\right]}{x^{2qm+2n+1}}=\frac{q^2x^{2q}}{(2qm+2n+q+1)(2qm+2n+2q+1)}$$

Hence we have

$$qS_q(x)=\sum_{n=0}^{q-1}\frac{x^{2n+1}}{\Gamma\left[\frac{2n+q+1}q\right]}{}_1F_2\left(1;\frac{2n+q+1}{2q},\frac{2n+2q+1}{2q};\frac14x^{2q}\right)$$

In the case that $q$ is even, less terms are needed:

$$2qS_{2q}(x)=\sum_{n=0}^{q-1}\frac{x^{2n+1}}{\Gamma\left[\frac{2n+2q+1}{2q}\right]}{}_1F_1\left(1;\frac{2n+2q+1}{2q};\frac12x^q\right)$$

Answer 2

Unfortunately, I don't understand the sense to make a sum of series out of one series. At least these sub-series should have something particular, not only another structure. Perhaps it's clearer to build a closed form using the incomplete gamma function.

$x!:=\Gamma(1+x) ~ ; ~~z>0$

$\displaystyle E_z(x):=\sum\limits_{k=0}^\infty\frac{x^k}{(kz)!}\hspace{2cm}$ Mittag-Leffler function

$\displaystyle f_z(x):=\sum\limits_{k=0}^\infty\frac{x^{k+z}}{(k+z)!}=e^x\left(1-\frac{\Gamma(z,x)}{\Gamma(z)}\right)$

$\displaystyle z:=\frac{1}{q} , ~ q\in\mathbb{N}$

$\displaystyle E_{1/q}(x) = e^{x^q} + \sum\limits_{j=1}^{q-1}f_{j/q}(x^q) = e^{x^q}\left(q - \sum\limits_{j=1}^{q-1}\frac{\Gamma(j/q,x^q)}{\Gamma(j/q)}\right) $

$\displaystyle S_q(x)=\frac{1}{q}\Im E_{1/q}(ix)$