relation to confluent hypergeometric functions

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1: 6.11 Relations to Other Functions

Confluent Hypergeometric Function

6.11.2

E_{1}\left(z\right)=e^{-z}U\left(1,1,z\right),

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Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral and $z$ : complex variable
Referenced by:: §6.11, 5th item, §6.6
Permalink:: http://dlmf.nist.gov/6.11.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §6.11, §6.11 and Ch.6

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6.11.3

\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)=U\left(1,1,-iz\right).

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Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\mathrm{i}$ : imaginary unit, $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals, $\mathrm{g}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals and $z$ : complex variable
Referenced by:: §6.11, 5th item
Permalink:: http://dlmf.nist.gov/6.11.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §6.11, §6.11 and Ch.6

2: 13.18 Relations to Other Functions

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§13.18(ii) Incomplete Gamma Functions

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§13.18(iii) Modified Bessel Functions

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§13.18(iv) Parabolic Cylinder Functions

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§13.18(v) Orthogonal Polynomials

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Laguerre Polynomials

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3: 13.6 Relations to Other Functions

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§13.6(ii) Incomplete Gamma Functions

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§13.6(iii) Modified Bessel Functions

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§13.6(v) Orthogonal Polynomials

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Laguerre Polynomials

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Charlier Polynomials

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4: 7.11 Relations to Other Functions

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Confluent Hypergeometric Functions

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5: 10.16 Relations to Other Functions

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Confluent Hypergeometric Functions

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6: 8.5 Confluent Hypergeometric Representations

§8.5 Confluent Hypergeometric Representations

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7: 10.39 Relations to Other Functions

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Confluent Hypergeometric Functions

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8: 18.34 Bessel Polynomials

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§18.34(i) Definitions and Recurrence Relation

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18.34.1

y_{n}\left(x;a\right)={{}_{2}F_{0}}\left({-n,n+a-1\atop-};-\frac{x}{2}\right)=% {\left(n+a-1\right)_{n}}\left(\frac{x}{2}\right)^{n}{{}_{1}F_{1}}\left({-n% \atop-2n-a+2};\frac{2}{x}\right).

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Defines:: $y_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Bessel polynomial
Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ : $=M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ notation for the Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.34(i)
Permalink:: http://dlmf.nist.gov/18.34.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §18.34(i), §18.34 and Ch.18

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9: 33.2 Definitions and Basic Properties

… тЦ║The function

F_{\ell}\left(\eta,\rho\right)

is recessive (§2.7(iii)) at

\rho=0

, and is defined by … тЦ║The functions

{H^{\pm}_{\ell}}\left(\eta,\rho\right)

are defined by …

10: 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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Laguerre

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