Kummer congruences

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1: 24.10 Arithmetic Properties

… ►Here and elsewhere two rational numbers are congruent if the modulus divides the numerator of their difference. ►

§24.10(ii) Kummer Congruences

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§24.10(iii) Voronoi’s Congruence

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2: 27.17 Other Applications

… ►Congruences are used in constructing perpetual calendars, splicing telephone cables, scheduling round-robin tournaments, devising systematic methods for storing computer files, and generating pseudorandom numbers. …Apostol and Zuckerman (1951) uses congruences to construct magic squares. …

3: 13.12 Products

§13.12 Products

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13.12.1 $$M(a,b,z)M(-a,-b,-z)+\frac{a(a-b)z^2}{b^2(1-b^2)}M(1+a,2+b,z)M(1-a,2-b,-z)=1.$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function and $z$: complex variable

Referenced by:

§13.12, §13.25

Permalink:

http://dlmf.nist.gov/13.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.12 and Ch.13

… ►For integral representations, integrals, and series containing products of $M(a,b,z)$ and $U(a,b,z)$ see Erdélyi et al. (1953a, §6.15.3).

4: 27.20 Methods of Computation: Other Number-Theoretic Functions

… ►A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function $\tau \left(n\right)$, and the values can be checked by the congruence (27.14.20). …

5: 13.30 Tables

§13.30 Tables

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Žurina and Osipova (1964) tabulates $M(a,b,x)$ and $U(a,b,x)$ for $b=2$, $a=-0.98(.02)1.10$, $x=0(.01)4$, 7D or 7S.

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Slater (1960) tabulates $M(a,b,x)$ for $a=-1(.1)1$, $b=0.1(.1)1$, and $x=0.1(.1)10$, 7–9S; $M(a,b,1)$ for $a=-11(.2)2$ and $b=-4(.2)1$, 7D; the smallest positive $x$-zero of $M(a,b,x)$ for $a=-4(.1)-0.1$ and $b=0.1(.1)2.5$, 7D.

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Abramowitz and Stegun (1964, Chapter 13) tabulates $M(a,b,x)$ for $a=-1(.1)1$, $b=0.1(.1)1$, and $x=0.1(.1)1(1)10$, 8S. Also the smallest positive $x$-zero of $M(a,b,x)$ for $a=-1(.1)-0.1$ and $b=0.1(.1)1$, 7D.

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Zhang and Jin (1996, pp. 411–423) tabulates $M(a,b,x)$ and $U(a,b,x)$ for $a=-5(.5)5$, $b=0.5(.5)5$, and $x=0.1,1,5,10,20,30$, 8S (for $M(a,b,x)$) and 7S (for $U(a,b,x)$).

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6: 13.3 Recurrence Relations and Derivatives

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§13.3(i) Recurrence Relations

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13.3.7 $$U(a-1,b,z)+(b-2a-z)U(a,b,z)+a(a-b+1)U(a+1,b,z)=0,$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.4.15

Permalink:

http://dlmf.nist.gov/13.3.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.3(i), §13.3 and Ch.13

… ►Kummer’s differential equation (13.2.1) is equivalent to … ►

13.3.14 $$(a+1)zU(a+2,b+2,z)+(z-b)U(a+1,b+1,z)-U(a,b,z)=0.$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

Referenced by:

§13.3(i)

Permalink:

http://dlmf.nist.gov/13.3.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §13.3(i), §13.3 and Ch.13

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§13.3(ii) Differentiation Formulas

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7: 13.6 Relations to Other Functions

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§13.6(i) Elementary Functions

… ►When $b=2a$ the Kummer functions can be expressed as modified Bessel functions. … ►

§13.6(v) Orthogonal Polynomials

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§13.6(vi) Generalized Hypergeometric Functions

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

8: Bibliography T

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New congruences for the Bernoulli numbers. Math. Comp. 48 (177), pp. 341–350.

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External Links:

ISSN 0025-5718, FullText, MathReview (Wells Johnson), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Asymptotic expansions of Kummer hypergeometric functions with three asymptotic parameters a, b and z. Indagationes Mathematicae.

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External Links:

ISSN 0019-3577, Document, Link

Referenced by:

§13.8(iv).

Encodings:

BibTeX

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Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters. Integral Transforms Spec. Funct. 33 (1), pp. 16–31.

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External Links:

ISSN 1065-2469, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§13.8(iv).

Encodings:

BibTeX

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9: Bibliography P

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A Kummer-type transformation for a ${}_2{}^{}F_2^{}$ hypergeometric function. J. Comput. Appl. Math. 173 (2), pp. 379–382.

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External Links:

ISSN 0377-0427, Document, MathReview (P. Anandani), ZentralBlatt

Referenced by:

§16.6.

Encodings:

BibTeX

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Voronoi type congruences for Bernoulli numbers. In Voronoi’s Impact on Modern Science. Book I, P. Engel and H. Syta (Eds.),

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External Links:

ZentralBlatt

Referenced by:

§24.10(iii).

Encodings:

BibTeX

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10: 13.13 Addition and Multiplication Theorems

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§13.13(i) Addition Theorems for $M(a,b,z)$

►The function $M(a,b,x+y)$ has the following expansions: … ►

§13.13(ii) Addition Theorems for $U(a,b,z)$

►The function $U(a,b,x+y)$ has the following expansions: … ►

§13.13(iii) Multiplication Theorems for $M(a,b,z)$ and $U(a,b,z)$

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