# §10.23 Sums

## §10.23(i) Multiplication Theorem

If and the upper signs are taken, then the restriction on is unnecessary.

The restriction is unnecessary when and is an integer. Special cases are:

10.23.3

### ¶ Graf’s and Gegenbauer’s Addition Theorems

Define

10.23.6

the branches being continuous and chosen so that and as . If , are real and positive and , then and are real and nonnegative, and the geometrical relationship is shown in Figure 10.23.1.

Figure 10.23.1: Graf’s and Gegenbauer’s addition theorems.

### ¶ Partial Fractions

For expansions of products of Bessel functions of the first kind in partial fractions see Rogers (2005).

## §10.23(iii) Series Expansions of Arbitrary Functions

### ¶ Neumann’s Expansion

10.23.10,

where is the distance of the nearest singularity of the analytic function from ,

and is Neumann’s polynomial, defined by the generating function:

10.23.12.

is a polynomial of degree in and

10.23.13.

For the more general form of expansion

see Watson (1944, §16.13), and for further generalizations see Watson (1944, Chapter 16) and Erdélyi et al. (1953b, §7.10.1).

### ¶ Fourier–Bessel Expansion

Assume satisfies

10.23.18

and define

where is as in §10.21(i). If , then

provided that is of bounded variation (§1.4(v)) on an interval with . This result is proved in Watson (1944, Chapter 18) and further information is provided in this reference, including the behavior of the series near and .

As an example,

(Note that when the left-hand side is 1 and the right-hand side is 0.)

### ¶ Other Series Expansions

For other types of expansions of arbitrary functions in series of Bessel functions, see Watson (1944, Chapters 17–19) and Erdélyi et al. (1953b, §§ 7.10.2–7.10.4). See also Schäfke (1960, 1961b).

## §10.23(iv) Compendia

For collections of sums of series involving Bessel or Hankel functions see Erdélyi et al. (1953b, §7.15), Gradshteyn and Ryzhik (2000, §§8.51–8.53), Hansen (1975), Luke (1969b, §9.4), Prudnikov et al. (1986b, pp. 651–691 and 697–700), and Wheelon (1968, pp. 48–51).