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I have strong feeling that the function, $$ f_\alpha (x) = \sum_{n=0}^\infty \frac{x^n}{n!\Gamma(1+n\alpha)}, $$ is a known special function (here $\Gamma(x)$ is the usual extension of the factorial). Is this the case?

David Handelman
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Daniel Parry
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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 03:14
  • 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 03:25
  • 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 03:33
  • How about $\alpha \in (-1,0)?$ if I don't mind asking. – Daniel Parry Apr 10 '14 at 03:35
  • 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 03:49
  • For natural values of $\alpha=A\in\mathbb N$, it can be re-written in terms of generalized hypergeometric functions as $f_A(x)=~_0F_A\bigg(;\dfrac1A,\dfrac2A,\ldots,\dfrac{A-1}A,1;\dfrac x{A^A}\bigg)$. – Lucian Apr 10 '14 at 10:26
  • I was searching for this...seems to be realeted to the moment-generating function for the Weibull distribution – kjetil b halvorsen Apr 17 '17 at 14:50

2 Answers2

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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.

Alexandre Eremenko
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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!

Sergei
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