Assuming equation (18.2.8) with its initialization defines a set of OP’s, , the corresponding associated orthogonal polynomials of order are the as defined by shifting the index in the recurrence coefficients by adding a constant , functions of , say , being replaced by . The inequality , for is replaced by
18.30.1 | |||
. | |||
The constant is usually taken as a positive integer. However, if the recurrence coefficients are polynomial, or rational, functions of , polynomials of degree may be well defined for provided that Askey and Wimp (1984).
The order recurrence is initialized as
18.30.2 | ||||
and then for consecutive
18.30.3 | |||
Associated polynomials and the related corecursive polynomials appear in Ismail (2009, §§2.3, 2.6, and 2.10), where the relationship of OP’s to continued fractions is made evident. The lowest order monic versions of both of these appear in §18.2(x), (18.2.31) defining the associated monic polynomials, and (18.2.32) their closely related cousins the corecursive polynomials.
These are defined by
18.30.4 | |||
, | |||
where is given by (18.30.2) and (18.30.3), with , , and as in (18.9.2). Explicitly,
18.30.5 | |||
where the generalized hypergeometric function 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
18.30.6 | |||
. | |||
Explicitly,
18.30.7 | |||
in which 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
18.30.8 | ||||
and
18.30.9 | |||
. | |||
Orthogonality
18.30.10 | |||
, , or , , | |||
with weight function
18.30.11 | |||
For the confluent hypergeometric function see §13.2(i). These constraints guarantee that the orthogonality only involves the integral , as above.
In view of (18.22.8) the associated Meixner–Pollaczek polynomials are defined by the recurrence relation
18.30.16 | ||||
. | ||||
They can be expressed in terms of type 3 Pollaczek polynomials (which are also associated type 2 Pollaczek polynomials) by (18.35.10).
Orthogonality
18.30.17 | |||
or , | |||
with weight function
18.30.18 | |||
For Gauss’ hypergeometric function see (15.2.1).
The corecursive orthogonal polynomials, , these being linearly independent solutions of the recurrence for the , are defined as follows:
18.30.21 | ||||
and then, as per usual, then, for consecutive ,
18.30.22 | |||
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.
The are also referred to as the numerator polynomials, the then being the denominator polynomials, in that the -th approximant of the continued fraction, ,
18.30.23 | |||
is given by
18.30.24 | |||
and of (18.30.23) and (18.30.24) are, also, precisely those of (18.2.34) and (18.2.35), now expressed via the traditional, , , coefficients, rather than the monic, , , recursion coefficients.
The ratio , as defined here, thus provides the same statement of Markov’s Theorem, as in (18.2.9_5), but now in terms of differently obtained numerator and denominator polynomials. Namely, if the interval is bounded, then
18.30.25 | |||
. | |||
Ismail (2009, §2.6) discusses this in a different notation; also note the assumption that , made throughout that reference, Ismail (2009, p. 16).
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 in , namely, , is unity; for a note on this standardization, see §18.2(iii). The notations and are used here to distinguish the two sets of monic polynomials from the (traditional) polynomials and of the preceding subsection.
The zeroth order corecursive monic polynomials follow directly from the alternate initialization
18.30.28 | ||||
followed by use of the recursion of (18.30.27).
It is easily seen that , and then
18.30.29 | |||
follows by induction on . This being the relationship established in §18.2(x) following (18.2.32). The usage of §18.2(x), where the monic associated polynomials, there denoted , instead of , are referred to as the first associated such polynomials in §18.2(x), is now evident. The ratio is then the of (18.2.35), leading to Markov’s theorem as stated in (18.30.25).
More generally, the th corecursive monic polynomials (defined with the initialization of (18.30.28) followed by the recurrence of (18.30.27)) are related to the st monic associated polynomials by
18.30.30 | |||
See Ismail (2009, p. 46 ), where the th corecursive polynomial is also related to an appropriate continued fraction, given here as its th convergent,
18.30.31 | |||