§18.30 Associated OP’s

These are defined by

where $p_n(x;c)$ is given by (18.30.2) and (18.30.3), with $A_n$, $B_n$, and $C_n$ as in (18.9.2). Explicitly,

where the generalized hypergeometric function ${}_4{}^{}F_3^{}$ is defined by (16.2.1).

For corresponding corecursive associated Jacobi polynomials, corecursive associated polynomials being discussed in §18.30(vii), see Letessier (1995). For other results for associated Jacobi polynomials, see Wimp (1987) and Ismail and Masson (1991).

These are defined by

Explicitly,

in which $P_n\left(x\right)$ are the Legendre polynomials of Table 18.3.1.

For further results on associated Legendre polynomials see Chihara (1978, Chapter VI, §12).

The recursion relation for the associated Laguerre polynomials, see (18.30.2), (18.30.3) is

and

Orthogonality

with weight function

For the confluent hypergeometric function $U$ see §13.2(i). These constraints guarantee that the orthogonality only involves the integral $x\in [0,\mathrm{\infty })$, as above.

For other cases there may also be, in addition to a possible integral as in (18.30.10), a finite sum of discrete weights on the negative real $x$-axis each multiplied by the polynomial product evaluated at the corresponding values of $x$, as in (18.2.3).

The recursion relation for the associated Hermite polynomials, see (18.30.2), and (18.30.3), is

and

Orthogonality

with weight function

For the parabolic cylinder function $U$ see §12.2(i).

In view of (18.22.8) the associated Meixner–Pollaczek polynomials ${𝒫}_n^{\lambda }(x;\varphi ,c)$ are defined by the recurrence relation

They can be expressed in terms of type 3 Pollaczek polynomials (which are also associated type 2 Pollaczek polynomials) by (18.35.10).

Orthogonality

with weight function

For Gauss’ hypergeometric function $F$ see (15.2.1).

The results in the previous two subsections are special limits:

and

The corresponding results for $c=0$ appear as (18.21.12) and (18.21.13), respectively.

The corecursive orthogonal polynomials, $p_n^{(0)}(x)$, these being linearly independent solutions of the recurrence for the $p_n(x)$, are defined as follows:

and then, as per usual, then, for consecutive $n=1,2,\mathrm{\dots }$,

Note that this is the same recurrence as in (18.2.8) for the traditional OP’s, but with a different initialization. Ismail (2009, §2.3) discusses the meaning of linearly independent in this situation.

Defining associated orthogonal polynomials and their relationship to their corecursive counterparts is particularly simple via use of the recursion relations for the monic, rather than via those for the traditional polynomials. The simplicity of the relationship follows from the fact that the monic polynomials have been rescaled so that the coefficient of the highest power of $x$ in $p_n(x)$, namely, $x^n$, is unity; for a note on this standardization, see §18.2(iii). The notations ${\widehat{p}}_n(x;c)$ and ${\widehat{p}}_n^{(0)}(x)$ are used here to distinguish the two sets of monic polynomials from the (traditional) polynomials $p_n(x;c)$ and $p_n^{(0)}(x)$ of the preceding subsection.

For associated Askey–Wilson polynomials see Rahman (2001). The type 3 Pollaczek polynomials are the associated type 2 Pollaczek polynomials, see §18.35. The relationship (18.35.8) implies the definition for the associated ultraspherical polynomials $C_n^{(\lambda )}(x;c)=P_n^{(\lambda )}(x;0,0,c)$.