Does the series $\sum_{k=0}^\infty \frac{a^{nk}b^{mk}}{(nk)!(mk)!}$ have a closed-form solution?

Question

I am currently trying to find a closed form formula for this series:

$$\sum_{k=0}^\infty \frac{a^{nk}b^{mk}}{(nk)!(mk)!}$$

with $a,b \in \mathbb{R}^*_+$, $n,m \in \mathbb{N}^*$, $\gcd(n,m) = 1$. (Don't know if this info is relevant)

I tried to find first the closed form formula for the easier series:

$$\sum_{k=0}^\infty \frac{a^{nk}}{(nk)!}$$

And intuitively by generalizing: $\frac{e^x+e^{-x}}{2} = \sum_{k=0}^\infty \frac{x^{2k}}{(2k)!}$ I found: $\frac{1}{n}\sum_{k=0}^{n-1} e^{\omega^k x} = \sum_{k=0}^\infty \frac{x^{nk}}{(nk)!}$, with $\omega = e^{2i\pi/n}$.

Wondering if with all of this information it is possible to find a closed form formula for the first series.

There's not going to be a closed form (depending on your definition of closed form.) There might be a way to write it in terms of known special functions. For $m=1$ and $n=1$ we get a modified Bessel function of the first kind. — Jair Taylor, Jul 19 '22 at 20:44
Yes, but it should be a “last resourt”; use the FoxH function. Also, simplify by setting $(a^nb^m)^k=x^k$. If $m,n=2^c,c\in\Bbb N$, then use the Legendre duplication formula multiple times and use the hypergeometric function — Тyma Gaidash, Jul 19 '22 at 20:57
Another "renaming-style" formulation: $\frac1{2\pi i}\oint\exp(az^m+bz^{-n})\frac{dz}{z}$. — metamorphy, Jul 20 '22 at 03:24

score 1 · Answer 1 · edited Jul 20 '22 at 15:43

1

As @Tyma Gaidash wrote in the comment we can express with FoxH special function.

$$ \sum_{k=0}^{\infty} \frac{a^{nk} b^{mk}}{(nk)!(mk)!} = H_{1,3}^{1,1}\bigg(-a^n b^m\bigg|{(0,1) \atop (0,1),(0,m),(0,n)}\bigg) $$

Mathematica code:

Inactivate[ Sum[(a^(n k)*b^(m k))/(((n k)!)*((m k)!)), {k, 0, Infinity}] == FoxH[{{{0, 1}}, {}}, {{{0, 1}}, {{0, m}, {0, n}}}, -a^n b^m], Sum]

edited Jul 20 '22 at 15:43

Cameron Williams

29,432

answered Jul 20 '22 at 15:35

Mariusz Iwaniuk

3,931

More simply, it is a Fox-Wright function, but that function is not directly in Mathematica nor other softwares. – Тyma Gaidash Jul 20 '22 at 15:39
Try to avoid using images in answers if possible. We have people who are sight-impaired on this site, and images make it impossible for them to contribute meaningfully. – Cameron Williams Jul 20 '22 at 15:44
@TymaGaidash It's FoxH function not Fox-Wright function. Mathematica has implemented this special function it was introduced in version 12.3. – Mariusz Iwaniuk Jul 20 '22 at 20:38
@CameronWilliams. Thanks you for editing. Mathematica can't convert FoxH function as TeXformat, probably a bug in TeXForm function. – Mariusz Iwaniuk Jul 20 '22 at 20:56
@MariuszIwaniuk Fox-wright is a special case of FoxH, as seen in the photo and article – Тyma Gaidash Jul 20 '22 at 21:01

Does the series $\sum_{k=0}^\infty \frac{a^{nk}b^{mk}}{(nk)!(mk)!}$ have a closed-form solution?

1 Answers1