equivalent equation for contiguous functions

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1: 15.5 Derivatives and Contiguous Functions

… ►An equivalent equation to the hypergeometric differential equation (15.10.1) is …

2: 16.4 Argument Unity

… ►The characterizing properties (18.22.2), (18.22.10), (18.22.19), (18.22.20), and (18.26.14) of the Hahn and Wilson class polynomials are examples of the contiguous relations mentioned in the previous three paragraphs. ►Contiguous balanced series have parameters shifted by an integer but still balanced. … … ►See Bailey (1964, §§4.3(7) and 7.6(1)) for the transformation formulas and Wilson (1978) for contiguous relations. …

3: 18.2 General Orthogonal Polynomials

… ►The recurrence relations (18.2.10) can be equivalently written as … ►Between the systems

\{p_{n}(x)\}

and

\{q_{n}(x)\}

there are the contiguous relations … ►Because of (18.2.36) the OP’s

p_{n}(x)

are also called monic denominator polynomials and the OP’s

p_{n-1}^{(1)}(x)

, or, equivalently, the

p_{n}^{(0)}(x)

, are called the monic numerator polynomials. … ►If

v(s)

is the formal power series such that

v(u(t))=t

then a property equivalent to (18.2.45) with

c_{n}=1

is that … ►

Monotonic Weight Functions

…