constant term identities

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21: 13.6 Relations to Other Functions

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13.6.5 $$M(a,a+1,-z)={\mathrm{e}}^{-z}M(1,a+1,z)=a{z}^{-a}\gamma (a,z),$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function and $z$: complex variable

A&S Ref:

13.6.10

Permalink:

http://dlmf.nist.gov/13.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(ii), §13.6 and Ch.13

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13.6.6 $$U(a,a,z)=z^{1-a}U(1,2-a,z)=z^{1-a}{\mathrm{e}}^z{E}_a\left(z\right)={\mathrm{e}}^z\mathrm{\Gamma }(1-a,z).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{\Gamma }(a,z)$: incomplete gamma function and $z$: complex variable

A&S Ref:

13.6.28 (in different form)

Permalink:

http://dlmf.nist.gov/13.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(ii), §13.6 and Ch.13

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13.6.9 $$M(\nu +\frac{1}{2},2\nu +1,2z)=\mathrm{\Gamma }\left(1+\nu \right){\mathrm{e}}^z{\left(z/2\right)}^{-\nu }{I}_{\nu }\left(z\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind and $z$: complex variable

A&S Ref:

13.6.3

Referenced by:

§13.11, §7.6(ii)

Permalink:

http://dlmf.nist.gov/13.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(iii), §13.6 and Ch.13

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13.6.11_1 $$M(\nu +\frac{1}{2},2\nu +1+n,2z)=\mathrm{\Gamma }\left(\nu \right){\mathrm{e}}^z{\left(z/2\right)}^{-\nu }\sum _{k=0}^n\frac{{\left(-n\right)}_k{\left(2\nu \right)}_k(\nu +k)}{{\left(2\nu +1+n\right)}_{k}k!}{I}_{\nu +k}\left(z\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $n$: nonnegative integer and $z$: complex variable

Source:

Prudnikov et al. (1990, page 579, (7.11.1.6))

Referenced by:

§13.11, §13.6(iii), §13.6(iii), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/13.6.E11_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §13.6(iii), §13.6 and Ch.13

… ►For representations of Coulomb functions in terms of Kummer functions see (33.2.4), (33.2.8) and (33.14.5).

22: 35.7 Gaussian Hypergeometric Function of Matrix Argument

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35.7.2 $$P_{\nu }^{(\gamma ,\delta )}\left(𝐓\right)=\frac{{\mathrm{\Gamma }}_m\left(\gamma +\nu +\frac{1}{2}(m+1)\right)}{{\mathrm{\Gamma }}_m\left(\gamma +\frac{1}{2}(m+1)\right)}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-\nu ,\gamma +\delta +\nu +\frac{1}{2}(m+1)}{\gamma +\frac{1}{2}(m+1)};𝐓),$$ ; $\gamma ,\delta ,\nu \in ℂ$; $\mathrm{\Re }\left(\gamma \right)>-1$.

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Defines:

$P_{\nu }^{(\gamma ,\delta )}\left(𝐓\right)$: Jacobi function of matrix argument

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $ℂ$: complex plane, $\in$: element of, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, : less-than for matrices, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Permalink:

http://dlmf.nist.gov/35.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.7(i), §35.7(i), §35.7 and Ch.35

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35.7.7 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{c};𝐈)=\frac{{\mathrm{\Gamma }}_m\left(c\right){\mathrm{\Gamma }}_m\left(c-a-b\right)}{{\mathrm{\Gamma }}_m\left(c-a\right){\mathrm{\Gamma }}_m\left(c-b\right)},$$ $\mathrm{\Re }\left(c\right),\mathrm{\Re }\left(c-a-b\right)>\frac{1}{2}(m-1)$.

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $b$: complex variable, $c$: complex variable and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

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35.7.8 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{c};𝐓)=\frac{{\mathrm{\Gamma }}_m\left(c\right){\mathrm{\Gamma }}_m\left(c-a-b\right)}{{\mathrm{\Gamma }}_m\left(c-a\right){\mathrm{\Gamma }}_m\left(c-b\right)}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{a+b-c+\frac{1}{2}(m+1)};𝐈-𝐓),$$ ; $\frac{1}{2}(j+1)-a\in \mathbb{N}$ for some $j=1,\mathrm{\dots },m$; $\frac{1}{2}(j+1)-c\notin \mathbb{N}$ and $c-a-b-\frac{1}{2}(m-j)\notin \mathbb{N}$ for all $j=1,\mathrm{\dots },m$.

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\in$: element of, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\notin$: not an element of, $\mathbb{N}$: set of all positive integers, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $c$: complex variable, $j$: nonnegative integer, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Referenced by:

§35.7(ii), Erratum (V1.0.26) for Equation (35.7.8)

Permalink:

http://dlmf.nist.gov/35.7.E8

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.26):

Originally the constraint was written as $\mathrm{\Re }\left(c\right),\mathrm{\Re }\left(c-a-b\right)>\frac{1}{2}(m-1)$. The constraint has been replaced with ; $\frac{1}{2}(j+1)-a\in \mathbb{N}$ for some $j=1,\mathrm{\dots },m$; $\frac{1}{2}(j+1)-c\notin \mathbb{N}$ and $c-a-b-\frac{1}{2}(m-j)\notin \mathbb{N}$ for all $j=1,\mathrm{\dots },m$; for details see https://arxiv.org/abs/2002.05248.

See also:

Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

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35.7.11 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{c};𝐈-{\alpha }^{-1}𝐓)$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $c$: complex variable and $\alpha \to \mathrm{\infty }$: parameter

Permalink:

http://dlmf.nist.gov/35.7.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §35.7(iv), §35.7 and Ch.35

… ►These approximations are in terms of elementary functions. …

23: 21.7 Riemann Surfaces

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§21.7(ii) Fay’s Trisecant Identity

… ►For all $𝐳\in {ℂ}^g$, and all $P_1$, $P_2$, $P_3$, $P_4$ on $\mathrm{\Gamma }$, Fay’s identity is given by … ►where again all integration paths are identical for all components. … ►

§21.7(iii) Frobenius’ Identity

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