§18.18 Sums

Let $f(z)$ be analytic within an ellipse $E$ with foci $z=\pm 1$, and

Then

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

Alternatively, assume $f(x)$ is real and continuous and $f^{\prime }(x)$ is piecewise continuous on $(-1,1)$. Assume also the integrals ${\int }_{-1}^1{(f(x))}^2{(1-x)}^{\alpha }{(1+x)}^{\beta }dx$ and ${\int }_{-1}^1{(f^{\prime }(x))}^2{(1-x)}^{\alpha +1}{(1+x)}^{\beta +1}dx$ converge. Then (18.18.2), with $z$ replaced by $x$, applies when ; moreover, the convergence is uniform on any compact interval within $(-1,1)$.

See §3.11(ii), or set $\alpha =\beta =\pm \frac{1}{2}$ in the above results for Jacobi and refer to (18.7.3)–(18.7.6).

This is the case $\alpha =\beta =0$ of Jacobi. Equation (18.18.1) becomes

Assume $f(x)$ is real and continuous and $f^{\prime }(x)$ is piecewise continuous on $(0,\mathrm{\infty })$. Assume also ${\int }_0^{\mathrm{\infty }}{(f(x))}^2{\mathrm{e}}^{-x}{x}^{\alpha }dx$ converges. Then

where

The convergence of the series (18.18.4) is uniform on any compact interval in $(0,\mathrm{\infty })$.

Assume $f(x)$ is real and continuous and $f^{\prime }(x)$ is piecewise continuous on $(-\mathrm{\infty },\mathrm{\infty })$. Assume also ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{(f(x))}^2{\mathrm{e}}^{-x^2}dx$ converges. Then

where

The convergence of the series (18.18.6) is uniform on any compact interval in $(-\mathrm{\infty },\mathrm{\infty })$.

In all three cases of Jacobi, Laguerre and Hermite, if $f(x)$ is $L^2$ on the corresponding interval with respect to the corresponding weight function and if $a_n,b_n,d_n$ are given by (18.18.1), (18.18.5), (18.18.7), respectively, then the respective series expansions (18.18.2), (18.18.4), (18.18.6) are valid with the sums converging in $L^2$ sense. See Szegő (1975, Theorems 3.1.5 and 5.7.1). See also (18.2.24), (18.2.25).

For the case $\lambda =\frac{1}{2}$ use (18.18.9); compare (18.7.9).

For integral representations for products implied by (18.18.8) and (18.18.9) see (18.17.5) and (18.17.6), respectively. For (18.18.8) see also (14.30.9). For formulas for Jacobi and Laguerre polynomials analogous to (18.18.8) and (18.18.9), see (Koornwinder, 1975b, 1977).

and a similar pair of equations by symmetry; compare the second row in Table 18.6.1. See Andrews et al. (1999, Lemma 7.1.1) for the more general expansion of $P_n^{(\gamma ,\delta )}\left(x\right)$ in terms of $P_n^{(\alpha ,\beta )}\left(x\right)$.

See (18.5.11) for the limit case $\lambda \to 0$ of (18.18.16).

The coefficients in the expansions (18.18.22) and (18.18.23) are positive, provided that in the former case $\lambda >0$. See (18.17.45) and (18.17.49) for integrated forms of (18.18.22) and (18.18.23), respectively. See Rahman (1981) for the linearization formula for Jacobi polynomials and Zeng (1992) for the linearization coefficients for Laguerre polynomials.

With

For the modified Bessel function $I_{\nu }\left(z\right)$ see §10.25(ii). Formula (18.18.27) is known as the Hille–Hardy formula.

Formula (18.18.28) is known as the Mehler formula. See Ismail (2009, Theorem 4.7.2) for a formula called Kibble–Slepian formula, which generalizes (18.18.28).

These Poisson kernels are positive, provided that $x,y$ are real, , and in the case of (18.18.27) $x,y\ge 0$. For the Poisson kernel of Jacobi polynomials (the Bailey formula) see Bailey (1938).