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1: 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 ${\nabla }_y(\lambda (y))$. …

2: 18.19 Hahn Class: Definitions

… ►

Wilson class (or quadratic lattice class). These are OP’s $p_n(x)=p_n(\lambda (y))$ ($p_n(x)$ of degree $n$ in $x$, $\lambda (y)$ quadratic in $y$) where the role of the differentiation operator is played by $\frac{{\mathrm{\Delta }}_y}{{\mathrm{\Delta }}_y(\lambda (y))}$ or $\frac{{\nabla }_y}{{\nabla }_y(\lambda (y))}$ or $\frac{{\delta }_y}{{\delta }_y(\lambda (y))}$. The Wilson class consists of two discrete and two continuous families.

…

3: 26.2 Basic Definitions

… ►Given a finite set $S$ with permutation $\sigma$, a cycle is an ordered equivalence class of elements of $S$ where $j$ is equivalent to $k$ if there exists an $\mathrm{\ell }=\mathrm{\ell }(j,k)$ such that $j={\sigma }^{\mathrm{\ell }}(k)$, where ${\sigma }^1=\sigma$ and ${\sigma }^{\mathrm{\ell }}$ is the composition of $\sigma$ with ${\sigma }^{\mathrm{\ell }-1}$. …

4: 18.27 $q$-Hahn Class

… ►The $q$-Hahn class OP’s comprise systems of OP’s $\{p_n(x)\}$, $n=0,1,\mathrm{\dots },N$, or $n=0,1,2,\mathrm{\dots }$, that are eigenfunctions of a second order $q$-difference operator. …

5: 19.39 Software

… ►For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C14). … ►Unless otherwise stated, the functions are $K\left(k\right)$ and $E\left(k\right)$, with $0\le k^2(=m)\le 1$. …

6: 18.24 Hahn Class: Asymptotic Approximations

§18.24 Hahn Class: Asymptotic Approximations

… ►In particular, asymptotic formulas in terms of elementary functions are given when $z=x$ is real and fixed. … ►The first expansion holds uniformly for $\delta \le x\le 1+\delta$, and the second for $1-\delta \le x\le 1+{\delta }^{-1}$, $\delta$ being an arbitrary small positive constant. … ►Taken together, these expansions are uniformly valid for and for $a$ in unbounded intervals—each of which contains $[0,(1-\delta )n]$, where $\delta$ again denotes an arbitrary small positive constant. … ►Similar approximations are included for Jacobi, Krawtchouk, and Meixner polynomials.

7: 18.28 Askey–Wilson Class

§18.28 Askey–Wilson Class

… ► … ►

§18.28(ii) Askey–Wilson Polynomials

… ►

§18.28(x) Limit Relations

… ►Genest et al. (2016) showed that these polynomials coincide with the nonsymmetric Wilson polynomials in Groenevelt (2007).

8: 18.23 Hahn Class: Generating Functions

§18.23 Hahn Class: Generating Functions

… ►

Hahn

… ►

18.23.3 $${\left(1-\frac{1-p}{p}z\right)}^x{(1+z)}^{N-x}=\sum _{n=0}^N\left(\genfrac{}{}{0.0pt}{}{N}{n}\right)K_n(x;p,N)z^n,$$ $x=0,1,\mathrm{\dots },N$.

ⓘ

Symbols:

$K_n(x;p,N)$: Krawtchouk polynomial, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $N$: positive integer, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.2(xii), §18.23

Permalink:

http://dlmf.nist.gov/18.23.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.23, §18.23 and Ch.18

… ►

18.23.4 $${\left(1-\frac{z}{c}\right)}^x{(1-z)}^{-x-\beta }=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\beta \right)}_n}{n!}{M}_n(x;\beta ,c)z^n,$$ $x=0,1,2,\mathrm{\dots }$, .

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Symbols:

$M_n(x;\beta ,c)$: Meixner polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.2(xii)

Permalink:

http://dlmf.nist.gov/18.23.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.23, §18.23 and Ch.18

… ►

18.23.7 $${(1-{\mathrm{e}}^{\mathrm{i}\varphi }z)}^{-\lambda +\mathrm{i}x}{(1-{\mathrm{e}}^{-\mathrm{i}\varphi }z)}^{-\lambda -\mathrm{i}x}=\sum _{n=0}^{\mathrm{\infty }}{P}_n^{(\lambda )}(x;\varphi )z^n,$$ .

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Symbols:

$P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.2(xii), §18.23

Permalink:

http://dlmf.nist.gov/18.23.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.23, §18.23 and Ch.18

9: 18.20 Hahn Class: Explicit Representations

§18.20 Hahn Class: Explicit Representations

►

§18.20(i) Rodrigues Formulas

… ►For the Hahn polynomials $p_n(x)=Q_n(x;\alpha ,\beta ,N)$ and … ►

§18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions

… ►Here we use as convention for (16.2.1) with $b_q=-N$, $a_1=-n$, and $n=0,1,\mathrm{\dots },N$ that the summation on the right-hand side ends at $k=n$. …

10: 13.6 Relations to Other Functions

… ►

13.6.1 $$M(a,a,z)={\mathrm{e}}^z,$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

A&S Ref:

13.6.12

Referenced by:

§13.10(v)

Permalink:

http://dlmf.nist.gov/13.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(i), §13.6 and Ch.13

… ►

13.6.3 $$M(0,b,z)=U(0,b,z)=1,$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(i), §13.6 and Ch.13

… ►

13.6.5 $$M(a,a+1,-z)={\mathrm{e}}^{-z}M(1,a+1,z)=a{z}^{-a}\gamma (a,z),$$

ⓘ

Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function and $z$: complex variable

A&S Ref:

13.6.10

Permalink:

http://dlmf.nist.gov/13.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(ii), §13.6 and Ch.13

… ►When $b=2a$ the Kummer functions can be expressed as modified Bessel functions. … ►

Charlier Polynomials

…