Practice_2_generalised Schlömilch's series
Schlömilch's series is a kind of series defined by
It has a generalised definition which is given by Bessel function of the first kind of order
and Struve function of order .
Consider two series and
Denote ,
Obviously, these two series are generalised Schlömilch's series

Now let us evaluate their values.
It can be shown that
these two formulas hold when
Proof:
Firstly, let us calculate
The Mellin transform of Bessel function is given by
Hence, its inversion formula is given by
Plug the inversion formula into , and exchange the order of integration and summation.
where
Denote
Note that has simple poles at
Hence, residue theorem implies
Similarly, the other series can be evaluated via the same approach.
After a cumbersome but mechanical process, we can obtain