Kummer congruences

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21: 6.11 Relations to Other Functions

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6.11.2 $$E_1\left(z\right)={\mathrm{e}}^{-z}U(1,1,z),$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: 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 $$\mathrm{g}\left(z\right)+\mathrm{i}\mathrm{f}\left(z\right)=U(1,1,-\mathrm{i}z).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{i}$: imaginary unit, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: 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

22: 13.11 Series

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13.11.1 $$M(a,b,z)=\mathrm{\Gamma }\left(a-\frac{1}{2}\right){\mathrm{e}}^{\frac{1}{2}z}{\left(\frac{1}{4}z\right)}^{\frac{1}{2}-a}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(2a-1\right)}_s{\left(2a-b\right)}_s}{{\left(b\right)}_{s}s!}\left(a-\frac{1}{2}+s\right)I_{a-\frac{1}{2}+s}\left(\frac{1}{2}z\right),$$ $a+\frac{1}{2},b\ne 0,-1,-2,\mathrm{\dots }$,

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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, $s$: nonnegative integer and $z$: complex variable

Referenced by:

§13.6(iii), §13.6(iii), §8.7

Permalink:

http://dlmf.nist.gov/13.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.11 and Ch.13

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13.11.2 $$M(a,b,z)=\mathrm{\Gamma }\left(b-a-\frac{1}{2}\right){\mathrm{e}}^{\frac{1}{2}z}{\left(\frac{1}{4}z\right)}^{a-b+\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(2b-2a-1\right)}_s{\left(b-2a\right)}_s(b-a-\frac{1}{2}+s)}{{\left(b\right)}_{s}s!}{I}_{b-a-\frac{1}{2}+s}\left(\frac{1}{2}z\right),$$ $b-a+\frac{1}{2},b\ne 0,-1,-2,\mathrm{\dots }$.

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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, $s$: nonnegative integer and $z$: complex variable

Proof sketch:

Combine (13.24.1) with (13.14.4).

A&S Ref:

13.3.6

Referenced by:

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

Permalink:

http://dlmf.nist.gov/13.11.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §13.11 and Ch.13

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13.11.3 $$𝐌(a,b,z)={\mathrm{e}}^{\frac{1}{2}z}\sum _{s=0}^{\mathrm{\infty }}{A}_s{\left(b-2a\right)}^{\frac{1}{2}(1-b-s)}{\left(\frac{1}{2}z\right)}^{\frac{1}{2}(1-b+s)}{J}_{b-1+s}\left(\sqrt{2z(b-2a)}\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $s$: nonnegative integer, $z$: complex variable and $A_n$: coefficients

Source:

Slater (1960, §3.8)

A&S Ref:

13.3.7

Referenced by:

§13.11, Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/13.11.E3

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §13.11 and Ch.13

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$(n+1)A_{n+1}=(n+b-1)A_{n-1}+(2a-b)A_{n-2},$ $n=2,3,4,\mathrm{\dots }$.

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23: 13.31 Approximations

§13.31 Approximations

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§13.31(i) Chebyshev-Series Expansions

►Luke (1969b, pp. 35 and 25) provides Chebyshev-series expansions of $M(a,b,x)$ and $U(a,b,x)$ that include the intervals $0\le x\le \alpha$ and , respectively, where $\alpha$ is an arbitrary positive constant. … ►

13.31.3 $$z^{a}U(a,1+a-b,z)=\underset{n\to \mathrm{\infty }}{lim}\frac{A_n(z)}{B_n(z)}.$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $n$: nonnegative integer, $z$: complex variable, $A_n(z)$: expansion and $B_n(z)$: expansion

Permalink:

http://dlmf.nist.gov/13.31.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.31(iii), §13.31 and Ch.13

24: Bibliography K

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The congruences of Clausen-von Staudt and Kummer for Bernoulli-Hurwitz numbers. Math. Ann. 216 (1), pp. 1–4.

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External Links:

ISSN 0025-5831, Document, MathReview (M. Basmakov), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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25: 13.4 Integral Representations

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§13.4(i) Integrals Along the Real Line

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13.4.5 $$U(a,b,z)=\frac{z^{1-a}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(1+a-b\right)}{\int }_0^{\mathrm{\infty }}\frac{U(b-a,b,t){\mathrm{e}}^{-t}{t}^{a-1}}{t+z}dt,$$ , $\mathrm{\Re }a>\max (\mathrm{\Re }b-1,0)$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§13.10(vi), §13.4(i), (9.10.18)

Permalink:

http://dlmf.nist.gov/13.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(i), §13.4 and Ch.13

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§13.4(ii) Contour Integrals

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§13.4(iii) Mellin–Barnes Integrals

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26: 12.20 Approximations

… ►Luke (1969b, pp. 25 and 35) gives Chebyshev-series expansions for the confluent hypergeometric functions $U(a,b,x)$ and $M(a,b,x)$ (§13.2(i)) whose regions of validity include intervals with endpoints $x=\mathrm{\infty }$ and $x=0$, respectively. …

27: 27.9 Quadratic Characters

… ►If $p$ does not divide $n$, then $(n|p)$ has the value $1$ when the quadratic congruence $x^2\equiv n(modp)$ has a solution, and the value $-1$ when this congruence has no solution. …

28: 13.9 Zeros

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§13.9(i) Zeros of $M(a,b,z)$

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§13.9(ii) Zeros of $U(a,b,z)$

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29: 13.7 Asymptotic Expansions for Large Argument

§13.7 Asymptotic Expansions for Large Argument

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§13.7(ii) Error Bounds

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13.7.4 $$U(a,b,z)=z^{-a}\sum _{s=0}^{n-1}\frac{{\left(a\right)}_s{\left(a-b+1\right)}_s}{s!}{(-z)}^{-s}+{\epsilon }_n(z),$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer, $s$: nonnegative integer, $z$: complex variable and ${\epsilon }_n(z)$: function

Referenced by:

§13.2(i), §13.7(ii)

Permalink:

http://dlmf.nist.gov/13.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §13.7(ii), §13.7 and Ch.13

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§13.7(iii) Exponentially-Improved Expansion

… ►For extensions to hyperasymptotic expansions see Olde Daalhuis and Olver (1995a).

30: 7.11 Relations to Other 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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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

A&S Ref:

7.1.21

Permalink:

http://dlmf.nist.gov/7.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §7.11, §7.11 and Ch.7

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7.11.5 $$\mathrm{erfc}z=\frac{1}{\sqrt{\pi }}{\mathrm{e}}^{-z^2}U(\frac{1}{2},\frac{1}{2},z^2)=\frac{z}{\sqrt{\pi }}{\mathrm{e}}^{-z^2}U(1,\frac{3}{2},z^2).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

Referenced by:

§7.11

Permalink:

http://dlmf.nist.gov/7.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.11, §7.11 and Ch.7

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7.11.6 $$C\left(z\right)+\mathrm{i}S\left(z\right)=zM(\frac{1}{2},\frac{3}{2},\frac{1}{2}\pi \mathrm{i}{z}^2)=z{\mathrm{e}}^{\pi \mathrm{i}{z}^2/2}M(1,\frac{3}{2},-\frac{1}{2}\pi \mathrm{i}{z}^2).$$

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Symbols:

$C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Permalink:

http://dlmf.nist.gov/7.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.11, §7.11 and Ch.7

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