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1: 16.2 Definition and Analytic Properties

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§16.2(i) Generalized Hypergeometric Series

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Polynomials

… ►Note also that any partial sum of the generalized hypergeometric series can be represented as a generalized hypergeometric function via … ►

§16.2(v) Behavior with Respect to Parameters

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2: 35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8 Generalized Hypergeometric Functions of Matrix Argument

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§35.8(i) Definition

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Convergence Properties

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§35.8(iv) General Properties

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Confluence

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3: 1.16 Distributions

… ► … ► $\mathrm{\Lambda }:𝒟(I)\to ℂ$ is called a distribution, or generalized function, if it is a continuous linear functional on $𝒟(I)$, that is, it is a linear functional and for every ${\varphi }_n\to \varphi$ in $𝒟(I)$, … ►More generally, for $\alpha :[a,b]\to [-\mathrm{\infty },\mathrm{\infty }]$ a nondecreasing function the corresponding Lebesgue–Stieltjes measure ${\mu }_{\alpha }$ (see §1.4(v)) can be considered as a distribution: … ►More generally, if $\alpha (x)$ is an infinitely differentiable function, then … ►Friedman (1990) gives an overview of generalized functions and their relation to distributions. …

4: 8.19 Generalized Exponential Integral

§8.19 Generalized Exponential Integral

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§8.19(i) Definition and Integral Representations

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§8.19(ii) Graphics

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§8.19(vi) Relation to Confluent Hypergeometric Function

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§8.19(xi) Further Generalizations

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5: 15.2 Definitions and Analytical Properties

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§15.2(i) Gauss Series

… ►In general, $F(a,b;c;z)$ does not exist when $c=0,-1,-2,\mathrm{\dots }$. … … ►

§15.2(ii) Analytic Properties

… ►The same properties hold for $F(a,b;c;z)$, except that as a function of $c$, $F(a,b;c;z)$ in general has poles at $c=0,-1,-2,\mathrm{\dots }$. …

6: 8.21 Generalized Sine and Cosine Integrals

§8.21 Generalized Sine and Cosine Integrals

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Spherical-Bessel-Function Expansions

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§8.21(vii) Auxiliary Functions

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§8.21(viii) Asymptotic Expansions

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7: 17.1 Special Notation

§17.1 Special Notation

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$k,j,m,n,r,s$	nonnegative integers.
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… ► ►Another function notation used is the “idem” function: … ►Fine (1988) uses $F(a,b;t:q)$ for a particular specialization of a ${}_2{}^{}\varphi _1^{}$ function.

8: 35.7 Gaussian Hypergeometric Function of Matrix Argument

§35.7 Gaussian Hypergeometric Function of Matrix Argument

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§35.7(i) Definition

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Jacobi Form

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Confluent Form

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Integral Representation

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9: 35.6 Confluent Hypergeometric Functions of Matrix Argument

§35.6 Confluent Hypergeometric Functions of Matrix Argument

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§35.6(i) Definitions

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Laguerre Form

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§35.6(ii) Properties

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§35.6(iv) Asymptotic Approximations

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10: 15.10 Hypergeometric Differential Equation

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15.10.1 $$z(1-z)\frac{d^{2}w}{{dz}^2}+\left(c-(a+b+1)z\right)\frac{dw}{dz}-abw=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.11(i), §15.11(i), §15.17(ii), §15.19(ii), §15.2(ii), §15.5(ii), §16.13, §17.6(iv), §19.18(ii), §19.4(ii)

Permalink:

http://dlmf.nist.gov/15.10.E1

Encodings:

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See also:

Annotations for §15.10(i), §15.10 and Ch.15

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Singularity $z=0$

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Singularity $z=1$

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Singularity $z=\mathrm{\infty }$

… ►The $\left(\genfrac{}{}{0.0pt}{}{6}{3}\right)=20$ connection formulas for the principal branches of Kummer’s solutions are: …