generating functions

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1: 10.56 Generating Functions

§10.56 Generating Functions

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2: 10.12 Generating Function and Associated Series

§10.12 Generating Function and Associated Series

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3: 10.35 Generating Function and Associated Series

§10.35 Generating Function and Associated Series

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4: 18.12 Generating Functions

§18.12 Generating Functions

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Jacobi

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Ultraspherical

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Legendre

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Laguerre

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5: 27.4 Euler Products and Dirichlet Series

… ►The Riemann zeta function is the prototype of series of the form ►

27.4.4 $$F(s)=\sum _{n=1}^{\mathrm{\infty }}f(n)n^{-s},$$

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Symbols:

$n$: positive integer, $f$: multiplicative function and $F(s)$: generating function

Permalink:

http://dlmf.nist.gov/27.4.E4

Encodings:

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See also:

Annotations for §27.4 and Ch.27

… ►The function $F(s)$ is a generating function, or more precisely, a Dirichlet generating function, for the coefficients. The following examples have generating functions related to the zeta function: …

6: Howard S. Cohl

… ►Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and $q$-series. …

7: 26.5 Lattice Paths: Catalan Numbers

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§26.5(ii) Generating Function

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8: 14.21 Definitions and Basic Properties

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§14.21(iii) Properties

… ►The generating function expansions (14.7.19) (with $𝖯$ replaced by $P$) and (14.7.22) apply when ; (14.7.21) (with $𝖯$ replaced by $P$) applies when $|h|>\max \left|z\pm {\left(z^2-1\right)}^{1/2}\right|$.

9: 26.15 Permutations: Matrix Notation

… ►The rook polynomial is the generating function for $r_j(B)$: … ► $N(x,B)$ is the generating function: ►

26.15.6 $$N(x,B)=\sum _{k=0}^n{N}_k(B)x^k,$$

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Symbols:

$x$: real variable, $k$: nonnegative integer, $n$: nonnegative integer, $B$: subset and $N(x,B)$: generating function

Permalink:

http://dlmf.nist.gov/26.15.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §26.15 and Ch.26

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26.15.7 $$N(x,B)=\sum _{k=0}^n{r}_k(B)(n-k)!{(x-1)}^k.$$

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Symbols:

$!$: factorial (as in $n!$), $x$: real variable, $k$: nonnegative integer, $n$: nonnegative integer, $B$: subset, $r_j(B)$: number of rook positions and $N(x,B)$: generating function

Permalink:

http://dlmf.nist.gov/26.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §26.15 and Ch.26

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26.15.8 $$N_0(B)\equiv N(0,B)=\sum _{k=0}^n{(-1)}^k{r}_k(B)(n-k)!.$$

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Symbols:

$\equiv$: equals by definition, $!$: factorial (as in $n!$), $k$: nonnegative integer, $n$: nonnegative integer, $B$: subset, $r_j(B)$: number of rook positions and $N(x,B)$: generating function

Permalink:

http://dlmf.nist.gov/26.15.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §26.15 and Ch.26

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10: 26.6 Other Lattice Path Numbers

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§26.6(ii) Generating Functions

►For sufficiently small $|x|$ and $|y|$, … ►

26.6.6 $$\sum _{n=0}^{\mathrm{\infty }}D(n,n)x^n=\frac{1}{\sqrt{1-6x+x^2}},$$

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Symbols:

$n$: nonnegative integer, $D(m,n)$: Delannoy number and $x$: small

Permalink:

http://dlmf.nist.gov/26.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §26.6(ii), §26.6 and Ch.26

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26.6.7 $$\sum _{n=0}^{\mathrm{\infty }}M(n)x^n=\frac{1-x-\sqrt{1-2x-3x^2}}{2x^2},$$

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Symbols:

$n$: nonnegative integer, $M(n)$: Motzkin number and $x$: small

Referenced by:

§26.6(iii)

Permalink:

http://dlmf.nist.gov/26.6.E7

Encodings:

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See also:

Annotations for §26.6(ii), §26.6 and Ch.26

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26.6.8 $$\sum _{n,k=1}^{\mathrm{\infty }}N(n,k)x^n{y}^k=\frac{1-x-xy-\sqrt{{(1-x-xy)}^2-4x^{2}y}}{2x},$$

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Symbols:

$k$: nonnegative integer, $n$: nonnegative integer, $N(n,k)$: Narayana number, $x$: small and $y$: small

Permalink:

http://dlmf.nist.gov/26.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §26.6(ii), §26.6 and Ch.26

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