Hermite polynomial case

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21: 18.25 Wilson Class: Definitions

§18.25 Wilson Class: Definitions

… ►For the Wilson class OP’s $p_n(x)$ with $x=\lambda (y)$: if the $y$-orthogonality set is $\{0,1,\mathrm{\dots },N\}$, then the role of the differentiation operator $d/dx$ in the Jacobi, Laguerre, and Hermite cases is played by the operator ${\mathrm{\Delta }}_y$ followed by division by ${\mathrm{\Delta }}_y(\lambda (y))$, or by the operator ${\nabla }_y$ followed by division by ${\nabla }_y(\lambda (y))$. … ►

Further Constraints for Racah Polynomials

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§18.25(ii) Weights and Standardizations: Continuous Cases

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§18.25(iii) Weights and Normalizations: Discrete Cases

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22: 18.27 $q$-Hahn Class

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From Stieltjes–Wigert to Hermite

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§18.27(vii) Discrete $q$-Hermite I and II Polynomials

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Discrete $q$-Hermite I

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Discrete $q$-Hermite II

… ►For discrete $q$-Hermite II polynomials the measure is not uniquely determined. …

23: 32.10 Special Function Solutions

… ►In the case ${\epsilon }_1\alpha +{\epsilon }_2\beta =2$, the Riccati equation is … ►

§32.10(iv) Fourth Painlevé Equation

… ►In the case when $n=0$ in (32.10.15), the Riccati equation is …When $a+\frac{1}{2}$ is zero or a negative integer the $U$ parabolic cylinder functions reduce to Hermite polynomials (§18.3) times an exponential function; thus … ►In the case when $n=0$ in (32.10.23), the Riccati equation is …

24: 18.14 Inequalities

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Hermite

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Hermite

… ►except that when $\alpha =\beta =-\frac{1}{2}$ (Chebyshev case) $|P_n^{(\alpha ,\beta )}\left(x_{n,m}\right)|$ is constant. … ►

Hermite

►The successive maxima of $|H_n\left(x\right)|$ form a decreasing sequence for $x\le 0$, and an increasing sequence for $x\ge 0$. …

25: 19.14 Reduction of General Elliptic Integrals

… ►Cases in which can be included by application of (19.2.10). … ►There are four important special cases of (19.14.4)–(19.14.6), as follows. … ►(These four cases include 12 integrals in Abramowitz and Stegun (1964, p. 596).) ►

§19.14(ii) General Case

… ►It then improves the classical method by first applying Hermite reduction to (19.2.3) to arrive at integrands without multiple poles and uses implicit full partial-fraction decomposition and implicit root finding to minimize computing with algebraic extensions. …

26: 18.9 Recurrence Relations and Derivatives

… ►Formulas (18.9.5), (18.9.11), (18.9.13) are special cases of (18.2.16). Formulas (18.9.6), (18.9.12), (18.9.14) are special cases of (18.2.17). … ►

Jacobi

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Ultraspherical

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Hermite

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27: 18.38 Mathematical Applications

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Integrable Systems

… ►with $H_n\left(x\right)$ as in §18.3, satisfies the Toda equation … ►also the case $\beta =0$ of (18.14.26), was used in de Branges’ proof of the long-standing Bieberbach conjecture concerning univalent functions on the unit disk in the complex plane. … ►

Random Matrix Theory

►Hermite polynomials (and their Freud-weight analogs (§18.32)) play an important role in random matrix theory. …

28: 18.1 Notation

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Classical OP’s

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Hermite: $H_n\left(x\right)$, ${\mathrm{𝐻𝑒}}_n\left(x\right)$.

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Discrete $q$-Hermite I: $h_n(x;q)$.

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Discrete $q$-Hermite II: ${\tilde{h}}_n(x;q)$.

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Continuous $q$-Hermite: $H_n\left(x|q\right)$.

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29: 1.17 Integral and Series Representations of the Dirac Delta

… ►In this case … ►

Legendre Polynomials (§§14.7(i) and 18.3)

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Laguerre Polynomials (§18.3)

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Hermite Polynomials (§18.3)

… ►(1.17.22)–(1.17.24) are special cases of Morse and Feshbach (1953a, Eq. (6.3.11)). …

30: 18.2 General Orthogonal Polynomials

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Kernel Polynomials

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§18.2(vi) Zeros

… ►The generating functions (18.12.13), (18.12.15), (18.23.3), (18.23.4), (18.23.5) and (18.23.7) for Laguerre, Hermite, Krawtchouk, Meixner, Charlier and Meixner–Pollaczek polynomials, respectively, can be written in the form (18.2.45). In fact, these are the only OP’s which are Sheffer polynomials (with Krawtchouk polynomials being only a finite system) … ►Equations (18.14.3_5) and (18.14.8), both for $\alpha =0$, can be seen as special cases of this result for Jacobi and Laguerre polynomials, respectively.