leading coefficients

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21: 18.30 Associated OP’s

… ►Assuming equation (18.2.8) with its initialization defines a set of OP’s,

p_{n}(x)

, the corresponding associated orthogonal polynomials of order

c

are the

p_{n}(x;c)

as defined by shifting the index

n

in the recurrence coefficients by adding a constant

c

, functions of

n

, say

f(n)

, being replaced by

f(n+c)

. …However, if the recurrence coefficients are polynomial, or rational, functions of

n

, polynomials of degree

n

may be well defined for

c\in\mathbb{R}

provided that

A_{n+c}B_{n+c}\neq 0,n=0,1,\dots

Askey and Wimp (1984). … ►

F(z)

and

F_{n}(z)

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,

A_{n}

B_{n}

C_{n}

coefficients, rather than the monic,

\alpha_{n}

\beta_{n}

, recursion coefficients. … ►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

p_{n}(x)

, namely,

x^{n}

, is unity; for a note on this standardization, see §18.2(iii). … ►The ratio

\hat{p}_{n-1}(x;1)/\hat{p}_{n}(x)

is then the

F_{n}(x)

of (18.2.35), leading to Markov’s theorem as stated in (18.30.25). …