Chu–Vandermonde sums (first and second)

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31: 26.21 Tables

§26.21 Tables

►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$ for $m$ up to 50 and $n$ up to 25; extends Table 26.4.1 to $n=10$; tabulates Stirling numbers of the first and second kinds, $s(n,k)$ and $S(n,k)$, for $n$ up to 25 and $k$ up to $n$; tabulates partitions $p\left(n\right)$ and partitions into distinct parts $p(𝒟,n)$ for $n$ up to 500. … ►It also contains a table of Gaussian polynomials up to ${\left[\genfrac{}{}{0.0pt}{}{12}{6}\right]}_q$. …

32: 10.60 Sums

§10.60 Sums

►

§10.60(i) Addition Theorems

… ►

§10.60(ii) Duplication Formulas

… ►For further sums of series of spherical Bessel functions, or modified spherical Bessel functions, see §6.10(ii), Luke (1969b, pp. 55–58), Vavreck and Thompson (1984), Harris (2000), and Rottbrand (2000). ►

§10.60(iv) Compendia

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33: 10.41 Asymptotic Expansions for Large Order

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10.41.3 $$I_{\nu }\left(\nu z\right)\sim \frac{{\mathrm{e}}^{\nu \eta }}{{(2\pi \nu )}^{\frac{1}{2}}{(1+z^2)}^{\frac{1}{4}}}\sum _{k=0}^{\mathrm{\infty }}\frac{U_k(p)}{{\nu }^k},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\eta$ and $U_k(p)$: polynomial coefficient

A&S Ref:

9.7.7

Referenced by:

§10.41(iii), §10.41(iv)

Permalink:

http://dlmf.nist.gov/10.41.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.41(ii), §10.41 and Ch.10

►

10.41.4 $$K_{\nu }\left(\nu z\right)\sim {\left(\frac{\pi }{2\nu }\right)}^{\frac{1}{2}}\frac{{\mathrm{e}}^{-\nu \eta }}{{(1+z^2)}^{\frac{1}{4}}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{U_k(p)}{{\nu }^k},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\eta$ and $U_k(p)$: polynomial coefficient

A&S Ref:

9.7.8

Permalink:

http://dlmf.nist.gov/10.41.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.41(ii), §10.41 and Ch.10

►

10.41.5 $$I_{\nu }^{\prime }\left(\nu z\right)\sim \frac{{(1+z^2)}^{\frac{1}{4}}{\mathrm{e}}^{\nu \eta }}{{(2\pi \nu )}^{\frac{1}{2}}z}\sum _{k=0}^{\mathrm{\infty }}\frac{V_k(p)}{{\nu }^k},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\eta$ and $V_k(p)$: polynomial coefficient

A&S Ref:

9.7.9

Referenced by:

§10.41(iv)

Permalink:

http://dlmf.nist.gov/10.41.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.41(ii), §10.41 and Ch.10

… ►

10.41.14 $$J_{\nu }\left(\nu z\right)={\left(\frac{4\zeta }{1-z^2}\right)}^{\frac{1}{4}}\left(\frac{\mathrm{Ai}\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{1}{3}}}\left(\sum _{k=0}^{\mathrm{\ell }}\frac{A_k(\zeta )}{{\nu }^{2k}}+O\left(\frac{1}{{\zeta }^{3\mathrm{\ell }+3}}\right)\right)+\frac{{\mathrm{Ai}}^{\prime }\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{5}{3}}}\left(\sum _{k=0}^{\mathrm{\ell }-1}\frac{B_k(\zeta )}{{\nu }^{2k}}+O\left(\frac{1}{{\zeta }^{3\mathrm{\ell }+1}}\right)\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $J_{\nu }\left(z\right)$: Bessel function of the first kind, $O\left(x\right)$: order not exceeding, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $A_k(\zeta )$: coefficients and $B_k(\zeta )$: coefficients

Referenced by:

§10.41(v)

Permalink:

http://dlmf.nist.gov/10.41.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §10.41(v), §10.41 and Ch.10

►

10.41.15 $$Y_{\nu }\left(\nu z\right)=-{\left(\frac{4\zeta }{1-z^2}\right)}^{\frac{1}{4}}\left(\frac{\mathrm{Bi}\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{1}{3}}}\left(\sum _{k=0}^{\mathrm{\ell }}\frac{A_k(\zeta )}{{\nu }^{2k}}+O\left(\frac{1}{{\zeta }^{3\mathrm{\ell }+3}}\right)\right)+\frac{{\mathrm{Bi}}^{\prime }\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{5}{3}}}\left(\sum _{k=0}^{\mathrm{\ell }-1}\frac{B_k(\zeta )}{{\nu }^{2k}}+O\left(\frac{1}{{\zeta }^{3\mathrm{\ell }+1}}\right)\right)\right),$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $O\left(x\right)$: order not exceeding, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $A_k(\zeta )$: coefficients and $B_k(\zeta )$: coefficients

Referenced by:

§10.41(v)

Permalink:

http://dlmf.nist.gov/10.41.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §10.41(v), §10.41 and Ch.10

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34: 10.26 Graphics

… ►

►►

Figure 10.26.1: $I_0\left(x\right)$, $I_1\left(x\right)$, $K_0\left(x\right)$, $K_1\left(x\right)$, $0\le x\le 3$. Magnify

►

►►

Figure 10.26.2: ${\mathrm{e}}^{-x}{I}_0\left(x\right)$, ${\mathrm{e}}^{-x}{I}_1\left(x\right)$, ${\mathrm{e}}^x{K}_0\left(x\right)$, ${\mathrm{e}}^x{K}_1\left(x\right)$, $0\le x\le 10$. Magnify

… ►

►►

Figure 10.26.7: ${\tilde{I}}_{1/2}\left(x\right)$, ${\tilde{K}}_{1/2}\left(x\right)$, $0.01\le x\le 3$. Magnify

►

►►

Figure 10.26.8: ${\tilde{I}}_1\left(x\right)$, ${\tilde{K}}_1\left(x\right)$, $0.01\le x\le 3$. Magnify

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►►

Figure 10.26.9: ${\tilde{I}}_5\left(x\right)$, ${\tilde{K}}_5\left(x\right)$, $0.01\le x\le 3$. Magnify

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35: Bibliography O

… ►

Summing one- and two-dimensional series related to the Euler series. J. Comput. Appl. Math. 98 (2), pp. 245–271.

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External Links:

ISSN 0377-0427, Document, MathReview (Boris Petrovich Osilenker), ZentralBlatt

Referenced by:

§25.8.

Encodings:

BibTeX

… ►

Hyperasymptotics for nonlinear ODEs. II. The first Painlevé equation and a second-order Riccati equation. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2062), pp. 3005–3021.

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External Links:

ISSN 1364-5021, Document, MathReview, ZentralBlatt

Referenced by:

§2.11(v), §32.12(i).

Encodings:

BibTeX

… ►

Algorithm 22: Riccati-Bessel functions of first and second kind. Comm. ACM 3 (11), pp. 600–601.

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Notes:

Includes Algol code, minimum accuracy 9S. For certification see same journal 13(1970), p. 448.

External Links:

ISSN 0001-0782, Document

Referenced by:

§10.77(ii).

Encodings:

BibTeX

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36: 10.56 Generating Functions

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10.56.1 $$\frac{\cos \sqrt{z^2-2zt}}{z}=\frac{\cos z}{z}+\sum _{n=1}^{\mathrm{\infty }}\frac{t^n}{n!}{𝗃}_{n-1}\left(z\right),$$

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Symbols:

$\cos z$: cosine function, $!$: factorial (as in $n!$), ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

Referenced by:

§10.56, Ch.10

Permalink:

http://dlmf.nist.gov/10.56.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.56 and Ch.10

►

10.56.2 $$\frac{\sin \sqrt{z^2-2zt}}{z}=\frac{\sin z}{z}+\sum _{n=1}^{\mathrm{\infty }}\frac{t^n}{n!}{𝗒}_{n-1}\left(z\right).$$

ⓘ

Symbols:

$!$: factorial (as in $n!$), $\sin z$: sine function, ${𝗒}_n\left(z\right)$: spherical Bessel function of the second kind, $n$: integer and $z$: complex variable

Referenced by:

§10.56

Permalink:

http://dlmf.nist.gov/10.56.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.56 and Ch.10

►

10.56.3 $$\frac{\cosh \sqrt{z^2+2\mathrm{i}zt}}{z}=\frac{\cosh z}{z}+\sum _{n=1}^{\mathrm{\infty }}\frac{{(\mathrm{i}t)}^n}{n!}{𝗂}_{n-1}^{(1)}\left(z\right),$$

ⓘ

Symbols:

$!$: factorial (as in $n!$), $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

Referenced by:

§10.56

Permalink:

http://dlmf.nist.gov/10.56.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.56 and Ch.10

►

10.56.4 $$\frac{\sinh \sqrt{z^2+2\mathrm{i}zt}}{z}=\frac{\sinh z}{z}+\sum _{n=1}^{\mathrm{\infty }}\frac{{(\mathrm{i}t)}^n}{n!}{𝗂}_{n-1}^{(2)}\left(z\right),$$

ⓘ

Symbols:

$!$: factorial (as in $n!$), $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, ${𝗂}_n^{(2)}\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

Referenced by:

§10.56

Permalink:

http://dlmf.nist.gov/10.56.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.56 and Ch.10

►

10.56.5 $$\frac{\exp \left(-\sqrt{z^2+2\mathrm{i}zt}\right)}{z}=\frac{{\mathrm{e}}^{-z}}{z}+\frac{2}{\pi }\sum _{n=1}^{\mathrm{\infty }}\frac{{(-\mathrm{i}t)}^n}{n!}{𝗄}_{n-1}\left(z\right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, ${𝗄}_n\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

Referenced by:

§10.56, Ch.10

Permalink:

http://dlmf.nist.gov/10.56.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.56 and Ch.10

37: 18.12 Generating Functions

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18.12.7 $$\frac{1-z^2}{1-2xz+z^2}=1+2\sum _{n=1}^{\mathrm{\infty }}{T}_n\left(x\right)z^n,$$ .

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

►

18.12.8 $$\frac{1-xz}{1-2xz+z^2}=\sum _{n=0}^{\mathrm{\infty }}{T}_n\left(x\right)z^n,$$ .

ⓘ

Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.9.6

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

►

18.12.9 $$-\ln \left(1-2xz+z^2\right)=2\sum _{n=1}^{\mathrm{\infty }}\frac{T_n\left(x\right)}{n}{z}^n,$$ .

ⓘ

Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $\ln z$: principal branch of logarithm function, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.9.8

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

►

18.12.10 $$\frac{1}{1-2xz+z^2}=\sum _{n=0}^{\mathrm{\infty }}{U}_n\left(x\right)z^n,$$ .

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Symbols:

$U_n\left(x\right)$: Chebyshev polynomial of the second kind, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.9.10

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.11 $$\frac{1}{\sqrt{1-2xz+z^2}}=\sum _{n=0}^{\mathrm{\infty }}{P}_n\left(x\right)z^n,$$ .

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.9.12

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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38: Bibliography V

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A Fortran computer program for calculating the oblate spheroidal radial functions of the first and second kind and their first derivatives. NRL Report No. 6959 Naval Res. Lab.  Washingtion, D.C..

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Referenced by:

§30.18(iii).

Encodings:

BibTeX

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Improved calculation of prolate spheroidal radial functions of the second kind and their first derivatives. Quart. Appl. Math. 62 (3), pp. 493–507.

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Notes:

For software, see Van Buren (website)

External Links:

ISSN 0033-569X, FullText, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§30.16(iii), §30.16(iii).

Encodings:

BibTeX

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On the series expansion method for computing incomplete elliptic integrals of the first and second kinds. Math. Comp. 23 (105), pp. 61–69.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§19.12, §19.5.

Encodings:

BibTeX

… ►

Modular hypergeometric residue sums of elliptic Selberg integrals. Lett. Math. Phys. 58 (3), pp. 223–238.

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External Links:

ISSN 0377-9017, Document, MathReview (Laurent Habsieger), ZentralBlatt

Referenced by:

§19.35(i).

Encodings:

BibTeX

… ►

Representation of an odd number as a sum of three primes (Russian). Dokl. Akad. Nauk SSSR 15, pp. 169–172 (Russian).

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External Links:

ZentralBlatt

Referenced by:

§27.13(ii).

Encodings:

BibTeX

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39: 14.13 Trigonometric Expansions

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14.13.1 $${𝖯}_{\nu }^{\mu }\left(\cos \theta \right)=\frac{2^{\mu +1}{(\sin \theta )}^{\mu }}{{\pi }^{1/2}}\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\nu +\mu +k+1\right)}{\mathrm{\Gamma }\left(\nu +k+\frac{3}{2}\right)}\frac{{\left(\mu +\frac{1}{2}\right)}_k}{k!}\sin \left((\nu +\mu +2k+1)\theta \right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\sin z$: sine function, $\mu$: general order, $\nu$: general degree and : variable

Source:

Volkmer (2021)

A&S Ref:

8.7.1

Referenced by:

§14.13, Erratum (V1.1.3) for Equations (14.13.1), (14.13.2)

Permalink:

http://dlmf.nist.gov/14.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.13 and Ch.14

►

14.13.2 $${𝖰}_{\nu }^{\mu }\left(\cos \theta \right)={\pi }^{1/2}{2}^{\mu }{(\sin \theta )}^{\mu }\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\nu +\mu +k+1\right)}{\mathrm{\Gamma }\left(\nu +k+\frac{3}{2}\right)}\frac{{\left(\mu +\frac{1}{2}\right)}_k}{k!}\cos \left((\nu +\mu +2k+1)\theta \right).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\sin z$: sine function, $\mu$: general order, $\nu$: general degree and : variable

Source:

Cohl et al. (2021, Theorem 6.2)

A&S Ref:

8.7.2

Referenced by:

§14.13, Erratum (V1.1.3) for Equations (14.13.1), (14.13.2)

Permalink:

http://dlmf.nist.gov/14.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.13 and Ch.14

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14.13.3 $${𝖯}_n\left(\cos \theta \right)=\frac{2^{2n+2}{(n!)}^2}{\pi (2n+1)!}\sum _{k=0}^{\mathrm{\infty }}\frac{1\cdot 3\mathrm{\cdots }(2k-1)}{k!}\frac{(n+1)(n+2)\mathrm{\cdots }(n+k)}{(2n+3)(2n+5)\mathrm{\cdots }(2n+2k+1)}\sin \left((n+2k+1)\theta \right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $\sin z$: sine function, $n$: nonnegative integer and : variable

A&S Ref:

8.7.3

Permalink:

http://dlmf.nist.gov/14.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.13 and Ch.14

►

14.13.4 $${𝖰}_n\left(\cos \theta \right)=\frac{2^{2n+1}{(n!)}^2}{(2n+1)!}\sum _{k=0}^{\mathrm{\infty }}\frac{1\cdot 3\mathrm{\cdots }(2k-1)}{k!}\frac{(n+1)(n+2)\mathrm{\cdots }(n+k)}{(2n+3)(2n+5)\mathrm{\cdots }(2n+2k+1)}\cos \left((n+2k+1)\theta \right),$$

ⓘ

Symbols:

$\cos z$: cosine function, $!$: factorial (as in $n!$), ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $n$: nonnegative integer and : variable

A&S Ref:

8.7.4

Permalink:

http://dlmf.nist.gov/14.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.13 and Ch.14

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40: 4.27 Sums

§4.27 Sums

►For sums of trigonometric and inverse trigonometric functions see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §§14–42), Oberhettinger (1973), and Prudnikov et al. (1986a, Chapter 5).