relation to generalized hypergeometric functions

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11: 13.6 Relations to Other Functions

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Charlier Polynomials

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§13.6(vi) Generalized Hypergeometric Functions

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12: 16.4 Argument Unity

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

13: 18.5 Explicit Representations

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

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Hermite

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14: 18.26 Wilson Class: Continued

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§18.26(i) Representations as Generalized Hypergeometric Functions and Dualities

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§18.26(iv) Generating Functions

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15: 8.19 Generalized Exponential Integral

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§8.19(vi) Relation to Confluent Hypergeometric Function

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16: 18.30 Associated OP’s

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§18.30(i) Associated Jacobi Polynomials

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17: 35.8 Generalized Hypergeometric Functions of Matrix Argument

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§35.8(ii) Relations to Other Functions

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18: 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. …

19: 15.9 Relations to Other Functions

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§15.9(i) Orthogonal Polynomials

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Jacobi

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Legendre

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Meixner

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Meixner–Pollaczek

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20: 18.23 Hahn Class: Generating Functions

§18.23 Hahn Class: Generating Functions

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

Hahn

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18.23.1 $${}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-x}{\alpha +1};-z){}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{x-N}{\beta +1};z)=\sum _{n=0}^N\frac{{\left(-N\right)}_n}{{\left(\beta +1\right)}_{n}n!}{Q}_n(x;\alpha ,\beta ,N)z^n,$$ $x=0,1,\mathrm{\dots },N$.

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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 $n!$), $N$: positive integer, $z$: complex variable, $n$: nonnegative 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

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18.23.2 $${}_2{}^{}F_0^{}(\genfrac{}{}{0pt}{}{-x,-x+\beta +N+1}{-};-z){}_2{}^{}F_0^{}(\genfrac{}{}{0pt}{}{x-N,x+\alpha +1}{-};z)=\sum _{n=0}^N\frac{{\left(-N\right)}_n{\left(\alpha +1\right)}_n}{n!}{Q}_n(x;\alpha ,\beta ,N)z^n,$$ $x=0,1,\mathrm{\dots },N$.

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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 ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $N$: positive integer, $z$: complex variable, $n$: nonnegative 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

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