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I am looking for a closed form expression for $$ f(x) = \sum_{k=0}^{\infty} \frac{\Gamma\left(a + \frac{k}{2}\right) x^k}{k!}, $$ where $2a$ is a positive integer, and $\left|x\right| \le 1$. I am guessing there is a connection to hypergeometric functions. I am not sure how to deal with the $k/2$ term in the $\Gamma$.

steveo'america
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  • Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. – José Carlos Santos Jan 24 '18 at 18:08
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    Maple evaluates this in terms of ${}_1F_1$ hypergeometrics. To deal with the $k/2$, take the even terms and the odd terms separately. – GEdgar Jan 24 '18 at 18:18
  • My effort so far was: 1. paging endlessly through Abramowitz & Stegun; 2. trying to replace the $x^k$ with $(x^2)^{k/2}$, but not getting far. – steveo'america Jan 24 '18 at 18:22

2 Answers2

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In terms of hypergeometric functions we have $$ \sum_{k\geq 0}\frac{\Gamma(a+k/2)x^k}{k!}=\Gamma(a)\cdot\phantom{ }_1 F_1\left(a;\tfrac{1}{2};\tfrac{x^2}{4}\right)+x\,\Gamma\left(a+\tfrac{1}{2}\right)\cdot\phantom{ }_1 F_1\left(a+\tfrac{1}{2};\tfrac{3}{2};\tfrac{x^2}{4}\right) $$ by just separating even/odd values of $k$. By the integral representation for the $\Gamma$ function and absolute convergence we are allowed to state $$ \sum_{k\geq 0}\frac{\Gamma(a+k/2)x^k}{k!}= \int_{0}^{+\infty}\sum_{k\geq 0}\frac{z^{a+k/2-1}x^k}{k!}e^{-z}\,dz=\int_{0}^{+\infty}z^{a-1}e^{x\sqrt{z}-z}\,dz$$ or, by enforcing the substitution $z\mapsto w^2$,

$$ \sum_{k\geq 0}\frac{\Gamma(a+k/2)x^k}{k!} = 2\int_{0}^{+\infty}w^{2a-1} e^{xw-w^2}\,dw$$ where the RHS can be estimated by applying Laplace's method centered at $w=\frac{x}{2}$, where $e^{xw-w^2}$ attains its maximum value. Since $2a$ is a positive integer, by applying integration by parts we may turn the RHS into a combination of polynomials, $e^{-x^2/4}$ and $\text{Erf}\left(\frac{x}{2}\right)$.

Jack D'Aurizio
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Actually there is a connection between the Parabolic Cylinder function:

Your function is summed up to

$$\large 2^{1-a} e^{\frac{x^2}{8}} \Gamma (2 a) D_{-2 a}\left(-\frac{x}{\sqrt{2}}\right)$$

More here

https://en.wikipedia.org/wiki/Parabolic_cylinder_function

Enrico M.
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