# Series of a Bessel function

## Main Question or Discussion Point

In solving a particular kind of integral I ended up with the following series

$$\sum_{k=0}^\infty \frac{\Gamma[b+k]}{\Gamma[a+b+k]} \frac{(1-t^2)^k}{k!} \left(\frac{\omega}{2}\right)^k J_{a+b-\frac{1}{2} +k} (\omega)$$

where 0<t<1, and a,b are small and positive.

I tried looking it up in a couple of books (Watson -- theory of Bessel functions, Prudnikov et al. -- Series and Integrals Vol 1-4, Gradshteyn -- Tables of Integrals) but this particular sum didn't appear in any of those (although some series came remarkably close). I tried substituting the bessel functions by a linear combination of Bessel functions times Lommel polynomials (see here) but this makes things even more complicated.

My question is, does anyone have either a good reference for a series like this, or knows some sort of method to solve it? Any hint is appreciated!

## Answers and Replies

Hello,

There is a book called "Integral representation and the computation of combinatorial sums" by G. P. Egorychev that might be useful. The general idea is to convert each term of the series to a contour integral and then using some theorems from several complex variables to manipulate the integrals before switching back to a series form. I have never used this approach myself so I am not sure how effective it is but it might be worth a try.

Good Luck!