Poisson equation

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11: 18.2 General Orthogonal Polynomials

… ►It is assumed throughout this chapter that for each polynomial

p_{n}(x)

that is orthogonal on an open interval

(a,b)

the variable

x

is confined to the closure of

(a,b)

unless indicated otherwise. (However, under appropriate conditions almost all equations given in the chapter can be continued analytically to various complex values of the variables.) … ►

Poisson kernel

►For OP’s

p_{n}

with

h_{n}

and orthogonality relation as in (18.2.5) and (18.2.5_5), the Poisson kernel is defined by …Instances where the Poisson kernel is nonnegative are of special interest, see Ismail (2009, Theorem 4.7.12). … ►Equations (18.14.3_5) and (18.14.8), both for

\alpha=0

, can be seen as special cases of this result for Jacobi and Laguerre polynomials, respectively.