discrete q-Hermite I and II polynomials

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21: 18.2 General Orthogonal Polynomials

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§18.2(ii) $x$-Difference Operators

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§18.2(vi) Zeros

… ►If $d\mu \in 𝐌(a,b)$ then the interval $[b-a,b+a]$ is included in the support of $d\mu$, and outside $[b-a,b+a]$ the measure $d\mu$ only has discrete mass points $x_k$ such that $b\pm a$ are the only possible limit points of the sequence $\{x_k\}$, see Máté et al. (1991, Theorem 10). … ►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) …

22: Bibliography F

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From continuous to discrete Painlevé equations. J. Math. Anal. Appl. 180 (2), pp. 342–360.

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External Links:

ISSN 0022-247X, Document, MathReview (Evangelos Ifantis), ZentralBlatt

Referenced by:

§32.7(ii).

Encodings:

BibTeX

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23: Bibliography N

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Error bounds and exponential improvement for Hermite’s asymptotic expansion for the gamma function. Appl. Anal. Discrete Math. 7 (1), pp. 161–179.

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External Links:

ISSN 1452-8630, Document, FullText, MathReview (Junesang Choi), ZentralBlatt

Referenced by:

§5.11(ii), §5.11(ii).

Encodings:

BibTeX

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24: 10.60 Sums

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10.60.3 $$\frac{{\mathrm{e}}^{-w}}{w}=\frac{2}{\pi }\sum _{n=0}^{\mathrm{\infty }}(2n+1){𝗂}_n^{(1)}\left(v\right){𝗄}_n\left(u\right)P_n\left(\cos \alpha \right),$$ .

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, ${𝗄}_n\left(z\right)$: modified spherical Bessel function and $n$: integer

A&S Ref:

10.2.35 (with condition added)

Referenced by:

§10.60(i)

Permalink:

http://dlmf.nist.gov/10.60.E3

Encodings:

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See also:

Annotations for §10.60(i), §10.60 and Ch.10

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10.60.8 $${\mathrm{e}}^{z\cos \alpha }=\sum _{n=0}^{\mathrm{\infty }}(2n+1){𝗂}_n^{(1)}\left(z\right)P_n\left(\cos \alpha \right),$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

A&S Ref:

10.2.36

Permalink:

http://dlmf.nist.gov/10.60.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.60(iii), §10.60 and Ch.10

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10.60.9 $${\mathrm{e}}^{-z\cos \alpha }=\sum _{n=0}^{\mathrm{\infty }}{(-1)}^n(2n+1){𝗂}_n^{(1)}\left(z\right)P_n\left(\cos \alpha \right).$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

A&S Ref:

10.2.37

Referenced by:

§10.60(iii), Ch.10

Permalink:

http://dlmf.nist.gov/10.60.E9

Encodings:

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See also:

Annotations for §10.60(iii), §10.60 and Ch.10

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25: 3.7 Ordinary Differential Equations

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§3.7(ii) Taylor-Series Method: Initial-Value Problems

… ►($𝐈$ and $\mathrm{𝟎}$ being the identity and zero matrices of order $2\times 2$.) … ►Let $(a,b)$ be a finite or infinite interval and $q(x)$ be a real-valued continuous (or piecewise continuous) function on the closure of $(a,b)$. … ►If $q(x)$ is $C^{\mathrm{\infty }}$ on the closure of $(a,b)$, then the discretized form (3.7.13) of the differential equation can be used. This converts the problem into a tridiagonal matrix problem in which the elements of the matrix are polynomials in $\lambda$; compare §3.2(vi). …

26: Bibliography G

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A new application of the discrete Laguerre polynomials in the numerical evaluation of the Hankel transform of a strongly decreasing even function. J. Comput. Phys. 42 (2), pp. 277–287.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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27: Bibliography S

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Some properties and expansions associated with the $q$-digamma function. Quaest. Math. 36 (1), pp. 67–77.

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External Links:

ISSN 1607-3606, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§5.18(ii), §5.18(ii).

Encodings:

BibTeX

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Time propagation of partial differential equations using the short iterative Lanczos method and finite-element discrete variable representation. Adv. Quantum Chem. 72, pp. 95–127.

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Referenced by:

§18.38(i).

Encodings:

BibTeX

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Relations among the fundamental solutions of the generalized hypergeometric equation when $p=q+1$. II. Logarithmic cases. Bull. Amer. Math. Soc. 45 (12), pp. 927–935.

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External Links:

Document, MathReview (C. S. Meijer), ZentralBlatt

Referenced by:

§16.8(ii).

Encodings:

BibTeX

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Uniform asymptotic expansions of Hermite polynomials. M. Phil. thesis, City University of Hong Kong.

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External Links:

Abstract

Referenced by:

§18.16(v).

Encodings:

BibTeX

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Orthogonal Polynomials. 4th edition, Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, RI.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§18.1(iii), §18.10(i), §18.10(ii), §18.10(iii), §18.11(ii), §18.12, §18.14(i), §18.14(iii), (18.15.4_5), §18.15(i), §18.15(ii), §18.15(ii), §18.15(ii), §18.15(iii), §18.15(iii), §18.15(iv), §18.15(v), §18.16(ii), §18.16(iv), §18.16(v), §18.16(vi), §18.17(viii), §18.18(i), §18.18(i), §18.18(viii), (18.2.1_5), (18.2.39), (18.2.4_5), §18.2(xi), §18.2(xii), §18.2(i), §18.2(iii), §18.2(iv), §18.2(v), Table 18.3.1, §18.31, §18.33(i), §18.33(ii), §18.33(iii), (18.35.9), §18.35(iii), §18.36(v), §18.5(i), §18.5(ii), §18.5(iii), §18.6(i), §18.7(i), §18.7(ii), §18.7(iii), §18.8, §18.9(i), §18.9(ii), §18.9(iii), Ch.18, Notations P.

Encodings:

BibTeX

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28: 8.7 Series Expansions

… ►For the functions $e_n(z)$, ${𝗂}_n^{(1)}\left(z\right)$, and $L_n^{(\alpha )}\left(x\right)$ see (8.4.11), §§10.47(ii), and 18.3, respectively. …

29: Bibliography W

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On the connection formulas of the third Painlevé transcendent. Discrete Contin. Dyn. Syst. 23 (1-2), pp. 541–560.

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External Links:

ISSN 1078-0947, Document, MathReview (Galina V. Filipuk), ZentralBlatt

Referenced by:

§32.13(ii).

Encodings:

BibTeX

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30: 10.49 Explicit Formulas

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10.49.8 $${𝗂}_n^{(1)}\left(z\right)=\frac{1}{2}{\mathrm{e}}^z\sum _{k=0}^n{(-1)}^k\frac{a_k(n+\frac{1}{2})}{z^{k+1}}+{(-1)}^{n+1}\frac{1}{2}{\mathrm{e}}^{-z}\sum _{k=0}^n\frac{a_k(n+\frac{1}{2})}{z^{k+1}}.$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $a_k(\nu )$: polynomial coefficient

A&S Ref:

10.2.9 (modified)

Referenced by:

§10.52(ii)

Permalink:

http://dlmf.nist.gov/10.49.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.49(ii), §10.49 and Ch.10

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10.49.10 $${𝗂}_n^{(2)}\left(z\right)=\frac{1}{2}{\mathrm{e}}^z\sum _{k=0}^n{(-1)}^k\frac{a_k(n+\frac{1}{2})}{z^{k+1}}+{(-1)}^n\frac{1}{2}{\mathrm{e}}^{-z}\sum _{k=0}^n\frac{a_k(n+\frac{1}{2})}{z^{k+1}}.$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, ${𝗂}_n^{(2)}\left(z\right)$: modified spherical Bessel function, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $a_k(\nu )$: polynomial coefficient

A&S Ref:

10.2.10 (modified)

Referenced by:

§10.52(ii)

Permalink:

http://dlmf.nist.gov/10.49.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.49(ii), §10.49 and Ch.10

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