power series in q

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11: 18.23 Hahn Class: Generating Functions

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18.23.1

{{}_{1}F_{1}}\left({-x\atop\alpha+1};-z\right){{}_{1}F_{1}}\left({x-N\atop% \beta+1};z\right)=\sum_{n=0}^{N}\frac{{\left(-N\right)_{n}}}{{\left(\beta+1% \right)_{n}}n!}Q_{n}\left(x;\alpha,\beta,N\right)z^{n},

x=0,1,\dots,N

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Symbols:: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, ${{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ : $=M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ notation for the Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $N$ : positive integer, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.23
Permalink:: http://dlmf.nist.gov/18.23.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §18.23, §18.23 and Ch.18

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18.23.2

{{}_{2}F_{0}}\left({-x,-x+\beta+N+1\atop-};-z\right)\*{{}_{2}F_{0}}\left({x-N,% x+\alpha+1\atop-};z\right)=\sum_{n=0}^{N}\frac{{\left(-N\right)_{n}}{\left(% \alpha+1\right)_{n}}}{n!}Q_{n}\left(x;\alpha,\beta,N\right)z^{n},

x=0,1,\dots,N

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Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $N$ : positive integer, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.23
Permalink:: http://dlmf.nist.gov/18.23.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §18.23, §18.23 and Ch.18

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18.23.4

\left(1-\frac{z}{c}\right)^{x}(1-z)^{-x-\beta}=\sum_{n=0}^{\infty}\frac{{\left% (\beta\right)_{n}}}{n!}M_{n}\left(x;\beta,c\right)z^{n},

x=0,1,2,\dots

|z|<1

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Symbols:: $M_{\NVar{n}}\left(\NVar{x};\NVar{\beta},\NVar{c}\right)$ : Meixner polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.2(xii)
Permalink:: http://dlmf.nist.gov/18.23.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §18.23, §18.23 and Ch.18

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18.23.5

{\mathrm{e}}^{z}\left(1-\frac{z}{a}\right)^{x}=\sum_{n=0}^{\infty}\frac{C_{n}% \left(x;a\right)}{n!}z^{n},

x=0,1,2,\dots

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Symbols:: $C_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Charlier polynomial, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.2(xii), §18.23
Permalink:: http://dlmf.nist.gov/18.23.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §18.23, §18.23 and Ch.18

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12: 16.11 Asymptotic Expansions

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§16.11(i) Formal Series

►For subsequent use we define two formal infinite series,

E_{p,q}(z)

and

H_{p,q}(z)

, as follows: … ►The formal series (16.11.2) for

H_{q+1,q}(z)

converges if

\left|z\right|>1

, and … ►Here the upper or lower signs are chosen according as

z

lies in the upper or lower half-plane; in consequence, in the fractional powers (§4.2(iv)) of

ze^{\mp\pi i}

its phases are

\operatorname{ph}z\mp\pi

, respectively. … ►The special case

a_{1}=1

p=q=2

is discussed in Kim (1972). …

13: 1.9 Calculus of a Complex Variable

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Powers

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§1.9(v) Infinite Sequences and Series

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§1.9(vi) Power Series

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Operations

… ►Lastly, a power series can be differentiated any number of times within its circle of convergence: …

14: 2.4 Contour Integrals

… ►Except that

\lambda

is now permitted to be complex, with

\Re\lambda>0

, we assume the same conditions on

q(t)

and also that the Laplace transform in (2.3.8) converges for all sufficiently large values of

\Re z

. … ►If

q(t)

is analytic in a sector

\alpha_{1}<\operatorname{ph}t<\alpha_{2}

containing

\operatorname{ph}t=0

, then the region of validity may be increased by rotation of the integration paths. … ►If, in addition, the corresponding integrals with

Q

and

F

replaced by their derivatives

Q^{(j)}

and

F^{(j)}

j=1,2,\dots,m

, converge uniformly, then by repeated integrations by parts … ►and apply the result of §2.4(iii) to each integral on the right-hand side, the role of the series (2.4.11) being played by the Taylor series of

p(t)

and

q(t)

t=t_{0}

. … ►in which

z

is a large real or complex parameter,

p(\alpha,t)

and

q(\alpha,t)

are analytic functions of

t

and continuous in

t

and a second parameter

\alpha

. …

15: 19.5 Maclaurin and Related Expansions

§19.5 Maclaurin and Related Expansions

… ► ►For Jacobi’s nome

q

: … ►Series expansions of

F\left(\phi,k\right)

and

E\left(\phi,k\right)

are surveyed and improved in Van de Vel (1969), and the case of

F\left(\phi,k\right)

is summarized in Gautschi (1975, §1.3.2). …

16: 3.11 Approximation Techniques

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§3.11(ii) Chebyshev-Series Expansions

… ►In fact, (3.11.11) is the Fourier-series expansion of

f(\cos\theta)

; compare (3.11.6) and §1.8(i). … ►For further details on Chebyshev-series expansions in the complex plane, see Mason and Handscomb (2003, §5.10). … ►be a formal power series. … ►When

F

has an explicit power-series expansion a possible choice of

R

is a Padé approximation to

F

. …

17: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions

§28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions

… ►With

{\cal C}_{\mu}^{(j)}

c^{\nu}_{n}(q)

A_{n}^{m}(q)

, and

B_{n}^{m}(q)

as in §28.23, …where

j=1,2,3,4

and

n\in\mathbb{Z}

. ►In the case when

\nu

is an integer, … ►For further power series of Mathieu radial functions of integer order for small parameters and improved convergence rate see Larsen et al. (2009).

18: 27.14 Unrestricted Partitions

… ►Multiplying the power series for

\mathit{f}\left(x\right)

with that for

1/\mathit{f}\left(x\right)

and equating coefficients, we obtain the recursion formula … ►Rademacher (1938) derives a convergent series that also provides an asymptotic expansion for

p\left(n\right)

: … ►This is related to the function

\mathit{f}\left(x\right)

in (27.14.2) by … ►Ono proved that for every prime

q>3

there are integers

a

and

b

such that

p\left(an+b\right)\equiv 0\pmod{q}

for all

n

. … ►The 24th power of

\eta\left(\tau\right)

in (27.14.12) with

e^{2\pi\mathrm{i}\tau}=x

is an infinite product that generates a power series in

x

with integer coefficients called Ramanujan’s tau function

\tau\left(n\right)

: …

19: 2.3 Integrals of a Real Variable

… ►converges for all sufficiently large

x

, and

q(t)

is infinitely differentiable in a neighborhood of the origin. … ►If, in addition,

q(t)

is infinitely differentiable on

[0,\infty)

and … ►For an extension with more general

t

-powers see Bleistein and Handelsman (1975, §4.1). … ►In addition to (2.3.7) assume that

f(t)

and

q(t)

are piecewise continuous (§1.4(ii)) on

(0,\infty)

, and … ►We now expand

f(\alpha,w)

in a Taylor series centered at the peak value

w=a

of the exponential factor in the integrand: …

20: Bibliography R

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RISC Combinatorics Group (website) Research Institute for Symbolic Computation, Hagenberg im Mühlkreis, Austria.

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Notes:: Software packages for hypergeometric and $q$ -hypergeometric summation, sequences and power series, permutation groups, partition analysis, and difference equations.
External Links:: RISC Combinatorics Group
Referenced by:: 6th item.
Encodings:: BibTeX

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H. Rosengren (2004) Elliptic hypergeometric series on root systems. Adv. Math. 181 (2), pp. 417–447.

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External Links:: ISSN 0001-8708, Document, MathReview, ZentralBlatt
Referenced by:: §20.11(iii).
Encodings:: BibTeX

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R. Roy (2011) Sources in the development of mathematics. Cambridge University Press, Cambridge.

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Notes:: Infinite Series and Products from the Fifteenth to the Twenty-first Century
External Links:: ISBN 978-0-521-11470-7, Document, Link, MathReview (Mehdi Hassani), ZentralBlatt
Referenced by:: Profile Ranjan Roy.
Encodings:: BibTeX

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W. Rudin (1973) Functional Analysis. McGraw-Hill Book Co., New York.

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Notes:: McGraw-Hill Series in Higher Mathematics
External Links:: MathReview (F. Smithies), ZentralBlatt
Referenced by:: §1.18(x), §2.6(i).
Encodings:: BibTeX

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W. Rudin (1976) Principles of Mathematical Analysis. 3rd edition, McGraw-Hill Book Co., New York.

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Notes:: International Series in Pure and Applied Mathematics
External Links:: MathReview, ZentralBlatt
Referenced by:: §1.4(iii).
Encodings:: BibTeX

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