Power series of ratio of Gamma functions

Question

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

Answer 1

There are closed-form expressions for all integer values of $a$; for $a=\{1,2,3,4\}$ these read _{$$\left\{\frac{1}{1-x},e^{x/4},\frac{1}{6\Gamma(1/3)}\left[3 x \, _1F_4\left(1;\frac{2}{3},\frac{5}{6},\frac{7}{6},\frac{4}{3};\frac{x^3}{11664}\right)+2\pi\ 3^{2/3} \left(\text{Bi}\left(\frac{\sqrt{x}}{\sqrt[3]{3}}\right)+\text{Bi}\left(-\frac{\sqrt{x}}{\sqrt[3]{3}}\right)\right)\right],\right.$$ $$\left.\, _0F_2\left(;\frac{1}{2},\frac{3}{4};\frac{x^2}{256}\right)+\frac{\Gamma \left(3/4\right)}{2\Gamma(1/4)}x \, _0F_2\left(;\frac{5}{4},\frac{3}{2};\frac{x^2}{256}\right)\right\}$$} these become more and more lengthy for larger integer $a$

Answer 2

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 $a>1$