Schl?milch'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?Schl?milch'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