Rogers?Dougall very well-poised sum

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1: 16.4 Argument Unity

… ►It is very well-poised if it is well-poised and $a_1=b_1+1$. … ►

Dixon’s Well-Poised Sum

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Rogers–Dougall Very Well-Poised Sum

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Dougall’s Very Well-Poised Sum

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2: 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

Referenced by:

§17.8

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:

pMML, png, TeX

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 well-poised when $r=s$ and … ►The series (17.4.1) is said to be very-well-poised when $r=s$, (17.4.11) is satisfied, and …

3: 17.9 Further Transformations of ${}_{r+1}{}^{}\varphi _r^{}$ Functions

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Bailey’s Transformation of Very-Well-Poised ${}_8{}^{}\varphi _7^{}$

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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

4: Bibliography M

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A $q$-analog of the ${}_5{}^{}F_4^{}(1)$ summation theorem for hypergeometric series well-poised in $\mathrm{𝑆𝑈}(n)$ . Adv. in Math. 57 (1), pp. 14–33.

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External Links:

ISSN 0001-8708, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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A $q$-analog of hypergeometric series well-poised in $\mathrm{𝑆𝑈}(n)$ and invariant $G$-functions. Adv. in Math. 58 (1), pp. 1–60.

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External Links:

ISSN 0001-8708, Document, MathReview (Robert A. Gustafson), ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.

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External Links:

Document, ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

BibTeX

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New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function. Proc. Nat. Acad. Sci. U.S.A. 93 (26), pp. 15004–15008.

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External Links:

ISSN 0027-8424, MathReview (Ken Ono), ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

BibTeX

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Nested sums, expansion of transcendental functions, and multiscale multiloop integrals. J. Math. Phys. 43 (6), pp. 3363–3386.

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External Links:

ISSN 0022-2488, Document, MathReview (Driss Essouabri), ZentralBlatt

Referenced by:

§16.24(ii).

Encodings:

BibTeX

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5: 27.19 Methods of Computation: Factorization

… ►The snfs can be applied only to numbers that are very close to a power of a very small base. …

6: 17.18 Methods of Computation

… ►Method (2) is very powerful when applicable (Andrews (1976, Chapter 5)); however, it is applicable only rarely. …

7: Diego Dominici

… ►Diego was very active in the SIAM activity group on Orthogonal Polynomials and Special Functions since 2010. …

8: 27.13 Functions

… ►The basic problem is that of expressing a given positive integer $n$ as a sum of integers from some prescribed set $S$ whose members are primes, squares, cubes, or other special integers. Each representation of $n$ as a sum of elements of $S$ is called a partition of $n$, and the number $S(n)$ of such partitions is often of great interest. … ►This conjecture dates back to 1742 and was undecided in 2009, although it has been confirmed numerically up to very large numbers. Vinogradov (1937) proves that every sufficiently large odd integer is the sum of three odd primes, and Chen (1966) shows that every sufficiently large even integer is the sum of a prime and a number with no more than two prime factors. … ►This problem is named after Edward Waring who, in 1770, stated without proof and with limited numerical evidence, that every positive integer $n$ is the sum of four squares, of nine cubes, of nineteen fourth powers, and so on. …

9: 37.21 Physical Applications

… ►A very important application is in lithography (see references in de Winter et al. (2020)), where Zernike polynomials are still used in the recent EUV (extreme ultraviolet) lithography developed by ASML and Zeiss companies. …

10: About MathML

… ►That format displays mathematics as static images and is neither very scalable nor accessible, but serves as a workable fallback. …