# Recognize this sum

I have strong feeling that the above function $$f_\alpha (x) = \sum_{n=0}^\infty \frac{x^n}{n!\Gamma(1+n\alpha)}$$ is a known special function but I can't seem to recognize it. Here $\Gamma(x)$ denotes the extension of the factorial. I want to know if anyone can recognize this.

Thanks

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Without the factorial, this would be the Mittag-Leffler function. Dividing on that factorial corresponds to a kind of Laplace transform. I don't think that Laplace transform of a Mittag-Leffler function is a "known function". – Alexandre Eremenko Apr 10 '14 at 3:14
You are right in that sense. I am wondering if there are other ways to identify this. The problem shows up in number theory my initial inclination is Bessel Functions. – Daniel Parry Apr 10 '14 at 3:25
When $\alpha>1$ is irrational, I can prove that it does not satisfy any linear ODE with polynomial coefficients. So it is unlikely to be related to Bessel or to any special function. – Alexandre Eremenko Apr 10 '14 at 3:33
How about $\alpha \in (-1,0)?$ if I don't mind asking. – Daniel Parry Apr 10 '14 at 3:35
If $\alpha<0$ my arguments are not valid, but then all depends on the arithmetic nature of alpha, how close its multiples can approximate integers. Still does not look like anything familiar. – Alexandre Eremenko Apr 10 '14 at 3:49

This is an entire function of order $1/\alpha$ when $\alpha>1$. So for irrational $\alpha$ it cannot satisfy any linear differential equation with polynomial coefficients. If $0<\alpha<1$, the order is $1$ but the type is minimal, so again it cannot satisfy any such equation. This excludes most special functions. (But does not exclude their compositions with some irrational power inside).

Entire solutions of linear differential equations with polynomial coefficients have rational order and normal type.

One can obtain an integral representation of this function by taking the integral representation of the Mittag-Leffler function and then a sort of Laplace transform of it.

Edit. If $\alpha=1$ it is expressed in terms of a Bessel function as the comment below shows.

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It is related to special functions! dlmf.nist.gov/10.46. Regardless, you helped me get in the right direction so I shall accept the answer anyway. – Daniel Parry Apr 10 '14 at 3:45
Thank you so much for your help! – Daniel Parry Apr 10 '14 at 3:46
The reference you gave shows relation to Bessel functions when $\alpha=1$ only. – Alexandre Eremenko Apr 10 '14 at 3:53
What do you mean by rational order and normal type? – joaopa Apr 10 '14 at 8:53
For the definition of order and type, see any book which has "Entire functions" in the title, the best one is Levin, Lectures on entire functions, or another book of the same author. – Alexandre Eremenko Apr 10 '14 at 13:47

This is so-called generalized Mittag-Leffler function, more exactly the Wright function (as series ) or the Fox function (as inverse Mellin transform). A lot is known about them

Start with 1. http://en.wikipedia.org/wiki/Fox%E2%80%93Wright_function You may use inet search with these names.

Other useful references are:

1. A.Kilbas, M.Saigo. H-transforms: theory and applications. http://books.google.ru/books?id=SL-HqdvUYzEC&pg=PA352&lpg=PA352&dq=Kilbas+saigo&source=bl&ots=xuQ-J79z6c&sig=RrYNqEKUIYv64RuK6urTemqUC28&hl=ru&sa=X&ei=eG5LU9bKEoa7ygOt4oCACQ&ved=0CD8Q6AEwAg#v=onepage&q=Kilbas%20saigo&f=false

2. A.M. Mathai, Ram Kishore Saxena, Hans J. Haubold. The H-Function: Theory and Applications. http://www.springer.com/physics/theoretical,+mathematical+%26+computational+physics/book/978-1-4419-0915-2

3. Papers of V.Kiryakova, e.g. Multiple (multiindex) Mittag–Leffler functions and relations to generalized fractional calculus: http://www.sciencedirect.com/science/article/pii/S0377042700002922 (and much more her papers on the subject).

4. MULTI-PARAMETRIC MITTAG-LEFFLER FUNCTIONS AND THEIR EXTENSION. Anatoly A. Kilbas , Anna A. Koroleva, Sergei V. Rogosin: http://link.springer.com/article/10.2478/s13540-013-0024-9

and so on... For sure you will find enough in these references, hope it will be useful!

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Thanks. These references will be quite useful! – Daniel Parry Apr 16 '14 at 4:57
Hey Sergi. Some work is coming out of this answer and I would like to try to give you an acknowledgement. Could you PM me your full name so you can get credit? – Daniel Parry Sep 26 '14 at 23:15
Thank you, but how to find your PM? – Sergei Sep 28 '14 at 19:37