# §18.18 Sums

## §18.18(i) Series Expansions of Arbitrary Functions

### ¶ Jacobi

Let be analytic within an ellipse with foci , and

Then

when lies in the interior of . Moreover, the series (18.18.2) converges uniformly on any compact domain within .

Alternatively, assume is real and continuous and is piecewise continuous on . Assume also the integrals and converge. Then (18.18.2), with replaced by , applies when ; moreover, the convergence is uniform on any compact interval within .

### ¶ Chebyshev

See §3.11(ii), or set in the above results for Jacobi and refer to (18.7.3)–(18.7.6).

### ¶ Laguerre

The convergence of the series (18.18.4) is uniform on any compact interval in .

### ¶ Hermite

Assume is real and continuous and is piecewise continuous on . Assume also converges. Then

where

The convergence of the series (18.18.6) is uniform on any compact interval in .

## §18.18(ii) Addition Theorems

### ¶ Legendre

For (18.18.8), (18.18.9), and the corresponding formula for Jacobi polynomials see Koornwinder (1975b). See also (14.30.9).

## §18.18(iv) Connection Formulas

### ¶ Jacobi

and a similar pair of equations by symmetry; compare the second row in Table 18.6.1.

## §18.18(v) Linearization Formulas

### ¶ Hermite

The coefficients in the expansions (18.18.22) and (18.18.23) are positive, provided that in the former case .

## §18.18(vii) Poisson Kernels

### ¶ Hermite

These Poisson kernels are positive, provided that are real, , and in the case of (18.18.27) .

## §18.18(viii) Other Sums

In this subsection the variables and are not confined to the closures of the intervals of orthogonality; compare §18.2(i).

## §18.18(ix) Compendia

For further sums see Hansen (1975, pp. 292-330), Gradshteyn and Ryzhik (2000, pp. 978–993), and Prudnikov et al. (1986b, pp. 637-644 and 700-718).