relation to generalized hypergeometric functions

SearchAdvancedHelp

(0.022 seconds)

11—20 of 69 matching pages

11: 18.26 Wilson Class: Continued

… ►

§18.26(i) Representations as Generalized Hypergeometric Functions

… ►

§18.26(iv) Generating Functions

…

12: 16.4 Argument Unity

… ►See Raynal (1979) for a statement in terms of $\mathit{3}j$ symbols (Chapter 34). … ► …

13: 13.6 Relations to Other Functions

… ►

Charlier Polynomials

… ►

§13.6(vi) Generalized Hypergeometric Functions

…

14: 18.5 Explicit Representations

… ►

§18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions

… ►

Hermite

…

15: 8.19 Generalized Exponential Integral

… ►

§8.19(vi) Relation to Confluent Hypergeometric Function

…

16: 35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►

§35.8(ii) Relations to Other Functions

…

17: 18.11 Relations to Other Functions

… ►See §§18.5(i) and 18.5(iii) for relations to trigonometric functions, the hypergeometric function, and generalized hypergeometric functions. …

18: 15.9 Relations to Other Functions

… ►

§15.9(i) Orthogonal Polynomials

… ►

Jacobi

… ►

Legendre

… ►

Meixner

… ►

Meixner–Pollaczek

…

19: 18.23 Hahn Class: Generating Functions

§18.23 Hahn Class: Generating Functions

►For the definition of generalized hypergeometric functions see §16.2. ►

Hahn

►

18.23.1 $${}_{1}F_1(\genfrac{}{}{0pt}{}{-x}{\alpha +1};-z){}_{1}F_1(\genfrac{}{}{0pt}{}{x-N}{\beta +1};z)=\sum _{n=0}^N\frac{{\left(-N\right)}_n}{{\left(\beta +1\right)}_{n}n!}{Q}_n(x;\alpha ,\beta ,N)z^n,$$ $x=0,1,\mathrm{\dots },N$.

ⓘ

Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, ${}_{1}F_1(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $z$: complex variable, $n$: nonnegative integer, $N$: positive integer and $x$: real variable

Referenced by:

§18.23

Permalink:

http://dlmf.nist.gov/18.23.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.23, §18.23 and Ch.18

►

18.23.2 $${}_{2}F_0(\genfrac{}{}{0pt}{}{-x,-x+\beta +N+1}{-};-z){}_{2}F_0(\genfrac{}{}{0pt}{}{x-N,x+\alpha +1}{-};z)=\sum _{n=0}^N\frac{{\left(-N\right)}_n{\left(\alpha +1\right)}_n}{n!}{Q}_n(x;\alpha ,\beta ,N)z^n,$$ $x=0,1,\mathrm{\dots },N$.

ⓘ

Symbols:

${}_{p}F_q(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_{p}F_q(\mathbf{a};\mathbf{b};z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{\mathbf{a}}{\mathbf{b}};z)$
generalized hypergeometric function, $Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $z$: complex variable, $n$: nonnegative integer, $N$: positive integer and $x$: real variable

Referenced by:

§18.23

Permalink:

http://dlmf.nist.gov/18.23.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.23, §18.23 and Ch.18

…

20: 16.25 Methods of Computation

§16.25 Methods of Computation

►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. They are similar to those described for confluent hypergeometric functions, and hypergeometric functions in §§13.29 and 15.19. There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations). …Instead a boundary-value problem needs to be formulated and solved. …