Let $a>1$ and define $G\_a(x)=\sum\limits\_{n=0}^{+\infty} \frac{\Gamma(\frac{2n+1}{a})}{\Gamma(2n+1)\Gamma(\frac{1}{a})}x^n$ where $\Gamma$ is the Gamma function. This series is convergent on $\mathbb{R}$ thanks to a ratio test and Stirling's formula.


For $a=2$, using Legendre duplication formula $\Gamma(z)\Gamma(z+\frac{1}{2}) = 2^{1-2z} \Gamma(\frac{1}{2})\Gamma(2z)$ with $z=n+\frac{1}{2}$ shows that $G\_2(x) = e^{\frac{x}{4}}$.


Can we get a somewhat explicit formula for other integer values of $a$, or even for arbitrary real values $a>1$? Has this series been studied somewhere?


For context, I am computing the Laplace transform of generalised Gaussian distributions [1], i.e their log-densities are $\propto -\lvert x\rvert^{a}$ (hence the case $a=2$ corresponds to the usual Gaussian measures). The above power series is related to the moment-generating function of such distributions.


I tried my luck a bit with hypergeometric functions and the multiplication theorem for the Gamma function (generalisation of the duplication formula) for small integer values of $a$ but did not quite make it.


References:
[1] <https://jsdajournal.springeropen.com/track/pdf/10.1186/s40488-018-0088-5.pdf>



Power series of ratio of Gamma functions
For arbitrary real $a > 0$ this is a special case of the generalized $\_p\Psi\_q(A;B;ζ)$ Fox-Wright function, where $A=[(a\_1,\alpha\_1),(a\_2,\alpha\_2),...,(a\_p,\alpha\_p)]$ and $B=[(b\_1,\beta\_1),(b\_2,\beta\_2),...,(b\_q,\beta\_q)]$ being $a\_j, j=1,..,p$ and $b\_k, k=1,..,q$ complex parameters and $\alpha\_j, \beta\_k$ are positive. $$\_p\Psi\_q(A;B;ζ)=\sum\_{n=0}^\infty \frac{ζ^n}{n!}\frac{\prod\_{j=1}^{p}\Gamma(a\_j+\alpha\_jn)}{\prod\_{k=1}^{q}\Gamma(b\_k+\beta\_kn)}$$ None gamma function in the numerator is singular. This means $$a\_{j}+α\_{j}m≠-ℓ,$$ with $j=1,2,..,p ∧ ℓ,m∈ℕ₀$. Series convergence depends on $\kappa, \rho, ϑ$ $$κ=∑\_{j=1}^{q}β\_{j}-∑\_{j=1}^{p}α\_{j}+1$$ $$ρ=∏\_{j=1}^{p}α\_{j}^{-α\_{j}}⋅∏\_{j=1}^{q}β\_{j}^{β\_{j}}$$ $$ϑ=½(q-p)+∑\_{j=1}^{p}a\_{j}-∑\_{j=1}^{q}b\_{j}$$ If $κ>0$ the series has an infinite radius of convergence and $\_p\Psi\_q(ζ)$ is an entire function. Series is uniformly and absolutely convergent for all finite $ζ$ . If $κ<0$ the sum is divergent for all nonzero values of $ζ$ whereas for $κ=0$ the function series has a finite radius of convergence $ρ$. Convergence on the boundary $|ζ|=ρ$ depends on parameter $ϑ$ converging absolutely if $ℜ(ϑ)<-½$.


For |arg$(-ζ)$$|<π-ε$, the Mellin-Barnes Integral $$\_{p}Ψ\_{q}(ζ)=\frac{1}{2πi}\int\_{L}\Gamma(s)⋅\frac{\prod\_{j=1}^{p}\Gamma(a\_{j}-α\_{j}s)}{\prod\_{k=1}^{q}\Gamma(b\_{k}-\beta\_{k}s)}(-ζ)^{-s}ds$$ defines a wider representation of Wright function. $L$ is a contour separating the poles of $\Gamma(s)$ to the left from those of $\Gamma(a\_{j}-α\_{j}s)$ to the right. For contour $L$ going from $-i\infty$ up to $+i\infty$ (possibly non-parallel to the vertical axis) this integral provides an analytical continuation of $\_{p}Ψ\_{q}(ζ)$ in $ζ∈ℂ\backslash [ρ,∞)$ when $ κ=0$.


This function is a special case of FoxH function, (See Wiki's or Wolfram's sites)
$$\_p\Psi\_q(A;B;ζ)=H\_{1+q,p}^{p,1}((1,1),B;A;-ζ^{-1})$$ For this particular case $A=[(1,1),(1/a,2/a)]$ and $B=[(1,2)]$. Thus $G\_a$ function is $$G\_a(x)=\frac{\_2\Psi\_1([(1,1),(1/a,2/a)];[(1,2)];x)}{\Gamma(1/a)}$$ $$G\_a(x)=\frac{H\_{2,2}^{2,1}([[(1,1)],[(1,2)]];[[(1,1),(1/a,2/a)],[\cdot]];-x^{-1})}{\Gamma(1/a)}$$ Note that $κ=2(1-1/a)$ and series converges for $a>1$ to an entire function. You can set this expression using FoxH function in Wolfram's Mathematica v13.0 in symbolic mode to see if there are some explicit formulae for other values of parameter $a$. I suggest try with $a\in \mathbb{Q}$ where $a>1$


