# relations to confluent hypergeometric functions and generalized hypergeometric functions

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##### 1: 7.11 Relations to Other Functions

###### §7.11 Relations to Other Functions

►###### Incomplete Gamma Functions and Generalized Exponential Integral

… ►###### Confluent Hypergeometric Functions

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7.11.4
$$\mathrm{erf}z=\frac{2z}{\sqrt{\pi}}M(\frac{1}{2},\frac{3}{2},-{z}^{2})=\frac{2z}{\sqrt{\pi}}{\mathrm{e}}^{-{z}^{2}}M(1,\frac{3}{2},{z}^{2}),$$

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###### Generalized Hypergeometric Functions

…##### 2: 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. …##### 3: 10.39 Relations to Other Functions

###### §10.39 Relations to Other Functions

… ►###### Parabolic Cylinder Functions

… ►###### Confluent Hypergeometric Functions

… ►For the functions $M$, $U$, ${M}_{0,\nu}$, and ${W}_{0,\nu}$ see §§13.2(i) and 13.14(i). ►###### Generalized Hypergeometric Functions and Hypergeometric Function

…##### 4: 10.16 Relations to Other Functions

###### §10.16 Relations to Other Functions

►###### Elementary Functions

… ►###### Confluent Hypergeometric Functions

… ►For the functions $M$ and $U$ see §13.2(i). … ►###### Generalized Hypergeometric Functions

…##### 5: 13.6 Relations to Other Functions

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###### Charlier Polynomials

…##### 6: 18.34 Bessel Polynomials

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###### §18.34(i) Definitions and Recurrence Relation

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18.34.1
$${y}_{n}(x;a)={}_{2}F_{0}(\genfrac{}{}{0pt}{}{-n,n+a-1}{-};-\frac{x}{2})={\left(n+a-1\right)}_{n}{\left(\frac{x}{2}\right)}^{n}{}_{1}F_{1}(\genfrac{}{}{0pt}{}{-n}{-2n-a+2};\frac{2}{x}).$$

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##### 7: 13.18 Relations to Other Functions

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###### §13.18(ii) Incomplete Gamma Functions

… ►###### §13.18(iv) Parabolic Cylinder Functions

… ►###### §13.18(v) Orthogonal Polynomials

… ►###### Hermite Polynomials

… ►###### Laguerre Polynomials

…##### 8: 16.18 Special Cases

###### §16.18 Special Cases

►The ${}_{1}F_{1}$ and ${}_{2}F_{1}$ functions introduced in Chapters 13 and 15, as well as the more general ${}_{p}F_{q}$ functions introduced in the present chapter, are all special cases of the Meijer $G$-function. This is a consequence of the following relations: …As a corollary, special cases of the ${}_{1}F_{1}$ and ${}_{2}F_{1}$ functions, including Airy functions, Bessel functions, parabolic cylinder functions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, are all special cases of the Meijer $G$-function. …##### 9: 18.20 Hahn Class: Explicit Representations

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###### §18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions

…##### 10: 18.5 Explicit Representations

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