numerator and denominator polynomials

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1: 18.30 Associated OP’s

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Numerator and Denominator Polynomials

►The $p_n^{(0)}(x)$ are also referred to as the numerator polynomials, the $p_n(x)$ then being the denominator polynomials, in that the $n$-th approximant of the continued fraction, $z\in ℂ$, … ►

Markov’s Theorem

►The ratio $p_n^{(0)}(z)/p_n(z)$, 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. …

2: 18.2 General Orthogonal Polynomials

… ►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. …

3: Errata

… ►We have significantly expanded the section on associated orthogonal polynomials, including expanded properties of associated Laguerre, Hermite, Meixner–Pollaczek, and corecursive orthogonal and numerator and denominator orthogonal polynomials. …

4: 18.39 Applications in the Physical Sciences

… ►The associated Coulomb–Laguerre polynomials are defined as …see Bethe and Salpeter (1957, p. 13), Pauling and Wilson (1985, pp. 130, 131); and noting that this differs from the Rodrigues formula of (18.5.5) for the Laguerre OP’s, in the omission of an $n!$ in the denominator. … ►Table 18.39.1 lists typical non-classical weight functions, many related to the non-classical Freud weights of §18.32, and §32.15, all of which require numerical computation of the recursion coefficients (i. … ►

The Coulomb–Pollaczek Polynomials

… ►As this follows from the three term recursion of (18.39.46) it is referred to as the J-Matrix approach, see (3.5.31), to single and multi-channel scattering numerics. …

5: 3.11 Approximation Techniques

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§3.11(i) Minimax Polynomial Approximations

… ►For the expansion (3.11.11), numerical values of the Chebyshev polynomials $T_n\left(x\right)$ can be generated by application of the recurrence relation (3.11.7). … ►Approximants with the same denominator degree are located in the same column of the table. … ►

Laplace Transform Inversion

►Numerical inversion of the Laplace transform (§1.14(iii)) …

6: 1.2 Elementary Algebra

… ►Let ${\alpha }_1,{\alpha }_2,\mathrm{\dots },{\alpha }_n$ be distinct constants, and $f(x)$ be a polynomial of degree less than $n$. … ►To find the polynomials $f_j(x)$, $j=1,2,\mathrm{\dots },n$, multiply both sides by the denominator of the left-hand side and equate coefficients. … ►Numerical methods and issues for solution of (1.2.61) appear in §§3.2(i) to 3.2(iii). … ►Eigenvalues are the roots of the polynomial equation …Numerical methods and issues for solution of (1.2.72) appear in §§3.2(iv) to 3.2(vii). …

7: 35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►The generalized hypergeometric function ${}_p{}^{}F_q^{}$ with matrix argument $𝐓\in 𝓢$, numerator parameters $a_1,\mathrm{\dots },a_p$, and denominator parameters $b_1,\mathrm{\dots },b_q$ is …