Pfaff–Saalschütz balanced sum

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1: 16.4 Argument Unity

… ►When $k=1$ the function is said to be balanced or Saalschützian. … ►

Pfaff–Saalschütz Balanced Sum

… ►Contiguous balanced series have parameters shifted by an integer but still balanced. … …

2: 35.8 Generalized Hypergeometric Functions of Matrix Argument

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35.8.1 $${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)=\sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}\sum _{|\kappa |=k}\frac{{\left[a_1\right]}_{\kappa }\mathrm{\cdots }{\left[a_p\right]}_{\kappa }}{{\left[b_1\right]}_{\kappa }\mathrm{\cdots }{\left[b_q\right]}_{\kappa }}{Z}_{\kappa }\left(𝐓\right).$$

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Defines:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument

Symbols:

$!$: factorial (as in $n!$), ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $k$: nonnegative integer, $p$: nonnegative integer, $q$: nonnegative integer and $\kappa$: vector of nonnegative integers

Referenced by:

§35.10, §35.10, §35.8(i), §35.8(i), §35.8(i), §35.8(i)

Permalink:

http://dlmf.nist.gov/35.8.E1

Encodings:

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See also:

Annotations for §35.8(i), §35.8 and Ch.35

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Pfaff–Saalschütz Formula

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3: 17.7 Special Cases of Higher ${}_r{}^{}\varphi _s^{}$ Functions

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Sum Related to (17.6.4)

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$q$-Pfaff–Saalschütz Sum

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Nonterminating Form of the $q$-Saalschütz Sum

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Gasper–Rahman $q$-Analogs of the Karlsson–Minton Sums

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Gosper’s Bibasic Sum

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4: 17.4 Basic Hypergeometric Functions

… ►In these references the factor ${\left({(-1)}^n{q}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}\right)}^{s-r}$ is not included in the sum. … ►

17.4.3 $${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,a_2,\mathrm{\dots },a_r}{b_1,b_2,\mathrm{\dots },b_s};q,z)={}_r{}^{}\psi _s^{}(a_1,a_2,\mathrm{\dots },a_r;b_1,b_2,\mathrm{\dots },b_s;q,z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(a_1,a_2,\mathrm{\dots },a_r;q)}_n{(-1)}^{(s-r)n}{q}^{(s-r)\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}{z}^n}{{(b_1,b_2,\mathrm{\dots },b_s;q)}_n}=\sum _{n=0}^{\mathrm{\infty }}\frac{{(a_1,a_2,\mathrm{\dots },a_r;q)}_n{(-1)}^{(s-r)n}{q}^{(s-r)\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}{z}^n}{{(b_1,b_2,\mathrm{\dots },b_s;q)}_n}+\sum _{n=1}^{\mathrm{\infty }}\frac{{(q/b_1,q/b_2,\mathrm{\dots },q/b_s;q)}_n}{{(q/a_1,q/a_2,\mathrm{\dots },q/a_r;q)}_n}{\left(\frac{b_1{b}_2\mathrm{\cdots }{b}_s}{a_1{a}_2\mathrm{\cdots }{a}_{r}z}\right)}^n.$$

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Defines:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $n$: nonnegative integer, $r$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §17.4(ii), §17.4 and Ch.17

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17.4.5 $${\mathrm{\Phi }}^{(1)}(a;b,b^{\prime };c;q;x,y)=\sum _{m,n\ge 0}\frac{{(a;q)}_{m+n}{(b;q)}_m{(b^{\prime };q)}_n{x}^m{y}^n}{{(q;q)}_m{(q;q)}_n{(c;q)}_{m+n}},$$

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Defines:

${\mathrm{\Phi }}^{(1)}(a;b,b^{\prime };c;q;x,y)$: first $q$-Appell function

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Referenced by:

Erratum (V1.0.10) for Section 17.1

Permalink:

http://dlmf.nist.gov/17.4.E5

Encodings:

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Correction (effective with 1.0.10):

The notation for ${\mathrm{\Phi }}^{(1)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument.

See also:

Annotations for §17.4(iii), §17.4 and Ch.17

… ►The series (17.4.1) is said to be balanced or Saalschützian when it terminates, $r=s$, $z=q$, and … ►The series (17.4.1) is said to be k-balanced when $r=s$ and …

5: Bibliography

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Pfaff’s method II: Diverse applications. J. Comput. Appl. Math. 68 (1-2), pp. 15–23.

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External Links:

ISSN 0377-0427, Document, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§17.7(iii).

Encodings:

BibTeX

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Theorems on generalized Dedekind sums. Pacific J. Math. 2 (1), pp. 1–9.

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External Links:

MathReview (N. G. de Bruijn), ZentralBlatt

Referenced by:

(25.11.41), (25.11.42), §25.11(xii).

Encodings:

BibTeX

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Bernoulli’s power-sum formulas revisited. Math. Gaz. 90 (518), pp. 276–279.

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Referenced by:

§24.4(iii).

Encodings:

BibTeX

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Positive Jacobi polynomial sums. II. Amer. J. Math. 98 (3), pp. 709–737.

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External Links:

FullText, MathReview (F. Schipp), ZentralBlatt

Referenced by:

§16.23(iii), Profile Richard A. Askey.

Encodings:

BibTeX

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Jacobi polynomials. I. New proofs of Koornwinder’s Laplace type integral representation and Bateman’s bilinear sum. SIAM J. Math. Anal. 5, pp. 119–124.

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External Links:

ISSN 1095-7154, Document, MathReview (R. P. Soni), ZentralBlatt

Referenced by:

§18.18(iv).

Encodings:

BibTeX

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6: 24.1 Special Notation

… ►It was used in Saalschütz (1893), Nielsen (1923), Schwatt (1962), and Whittaker and Watson (1927). …

7: Bibliography K

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The computation of the sums of negative even powers of roots of Bessel functions. Doklady Akad. Nauk SSSR (N.S.) 77, pp. 561–564.

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External Links:

MathReview (A. Erdélyi)

Referenced by:

§10.21(xiii).

Encodings:

BibTeX

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An extension of Saalschütz’s summation theorem for the series ${}_{r+3}{}^{}F_{r+2}^{}$ . Integral Transforms Spec. Funct. 24 (11), pp. 916–921.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview (Christian Lavault), ZentralBlatt

Referenced by:

§16.4(ii), §16.4(ii).

Encodings:

BibTeX

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Estimates of trigonometric sums and their applications. Uspehi Mat. Nauk 13 (4 (82)), pp. 185–192 (Russian).

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External Links:

MathReview (R. A. Rankin), ZentralBlatt

Referenced by:

§27.12.

Encodings:

BibTeX

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HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively $q$-binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.

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Notes:

Also published in J. Symbolic Comput. (1995), v. 20, no. 5–6, pp. 737–744.

External Links:

Code, Document, MathReview (G. Gasper), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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8: Bibliography S

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Vorlesungen über die Bernoullischen Zahlen, ihren Zusammenhang mit den Secanten-Coefficienten und ihre wichtigeren Anwendungen. Springer-Verlag, Berlin (German).

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External Links:

ZentralBlatt

Referenced by:

§24.1.

Encodings:

BibTeX

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Sums of certain series of the Riemann zeta function. J. Math. Anal. Appl. 134 (1), pp. 129–140.

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External Links:

ISSN 0022-247X, Document, MathReview (Aleksandar Ivić), ZentralBlatt

Referenced by:

(25.8.3), §25.8.

Encodings:

BibTeX

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9: 17.9 Further Transformations of ${}_{r+1}{}^{}\varphi _r^{}$ Functions

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Sears’ Balanced ${}_4{}^{}\varphi _3^{}$ Transformations

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§17.9(iv) Bibasic Series

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17.9.19 $$\sum _{n=0}^{\mathrm{\infty }}\frac{{(a;q^2)}_n{(b;q)}_n}{{(q^2;q^2)}_n{(c;q)}_n}{z}^n=\frac{{(b;q)}_{\mathrm{\infty }}{(az;q^2)}_{\mathrm{\infty }}}{{(c;q)}_{\mathrm{\infty }}{(z;q^2)}_{\mathrm{\infty }}}\sum _{n=0}^{\mathrm{\infty }}\frac{{(c/b;q)}_{2n}{(z;q^2)}_n{b}^{2n}}{{(q;q)}_{2n}{(az;q^2)}_n}+\frac{{(b;q)}_{\mathrm{\infty }}{(azq;q^2)}_{\mathrm{\infty }}}{{(c;q)}_{\mathrm{\infty }}{(zq;q^2)}_{\mathrm{\infty }}}\sum _{n=0}^{\mathrm{\infty }}\frac{{(c/b;q)}_{2n+1}{(zq;q^2)}_n{b}^{2n+1}}{{(q;q)}_{2n+1}{(azq;q^2)}_n}.$$

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.9.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §17.9(iv), §17.9(iv), §17.9 and Ch.17

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17.9.20 $$\sum _{n=0}^{\mathrm{\infty }}\frac{{(a;q^k)}_n{(b;q)}_{kn}{z}^n}{{(q^k;q^k)}_n{(c;q)}_{kn}}=\frac{{(b;q)}_{\mathrm{\infty }}{(az;q^k)}_{\mathrm{\infty }}}{{(c;q)}_{\mathrm{\infty }}{(z;q^k)}_{\mathrm{\infty }}}\sum _{n=0}^{\mathrm{\infty }}\frac{{(c/b;q)}_n{(z;q^k)}_n{b}^n}{{(q;q)}_n{(az;q^k)}_n},$$ $k=1,2,3,\mathrm{\dots }$.

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $k$: nonnegative integer, $q$: complex base, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.9.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §17.9(iv), §17.9(iv), §17.9 and Ch.17

10: 16.24 Physical Applications

… ►These are balanced ${}_4{}^{}F_3^{}$ functions with unit argument. …