# §28.23 Expansions in Series of Bessel Functions

We use the following notations:

compare §10.2(ii). For the coefficients see §28.14. For and see §28.4.

valid for all when , and for and when .

valid for all when , and for and when .

In the case when is an integer

When , each of the series (28.23.6)–(28.23.13) converges for all . When the series in the even-numbered equations converge for and , and the series in the odd-numbered equations converge for and .

For proofs and generalizations, see Meixner and Schäfke (1954, §§2.62 and 2.64).