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31: Errata

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The title of the paragraph which was previously “Andrews’ Terminating $q$-Analog of (17.7.8)” has been changed to “Andrews’ $q$-Analog of the Terminating Version of Watson’s ${}_3{}^{}F_2^{}$ Sum (16.4.6)”. The title of the paragraph which was previously “Andrews’ Terminating $q$-Analog” has been changed to “Andrews’ $q$-Analog of the Terminating Version of Whipple’s ${}_3{}^{}F_2^{}$ Sum (16.4.7)”.

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17.9.3 $${}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a,b}{c};q,z)=\frac{{(abz/c;q)}_{\mathrm{\infty }}}{{(bz/c;q)}_{\mathrm{\infty }}}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{a,c/b,0}{c,cq/(bz)};q,q)+\frac{{(a,bz,c/b;q)}_{\mathrm{\infty }}}{{(c,z,c/(bz);q)}_{\mathrm{\infty }}}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{z,abz/c,0}{bz,bzq/c};q,q)$$

Originally, the second term on the right-hand side was missing. The form of the equation where the second term is missing is correct if the ${}_2{}^{}\varphi _1^{}$ is terminating. It is this form which appeared in the first edition of Gasper and Rahman (1990). The more general version which appears now is what is reproduced in Gasper and Rahman (2004, (III.5)).

Reported by Roberto S. Costas-Santos on 2019-04-26

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32: 18.39 Applications in the Physical Sciences

… ►The recursion of (18.39.46) is that for the type 2 Pollaczek polynomials of (18.35.2), with $\lambda =l+1$, $a=-b=2Z/s$, and $c=0$, and terminates for $x=x_i^N$ being a zero of the polynomial of order $N$. …

33: 18.28 Askey–Wilson Class

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18.28.26 $$\underset{\lambda \to 0}{lim}{r}_n(x/(2\lambda );\lambda ,qa{\lambda }^{-1},qc{\lambda }^{-1},b{c}^{-1}\lambda |q)=P_n(x;a,b,c;q).$$

ⓘ

Symbols:

$P_n(x;a,b,c;q)$: big $q$-Jacobi polynomial, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $z$: complex variable, $q$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Source:

Koekoek et al. (2010, (14.1.18))

Proof sketch:

In the terminating $q$-hypergeometric series which defines the ${}_4{}^{}\varphi _3^{}$ in (18.28.1_5), with ${(az{q}^{\mathrm{\ell }},a{z}^{-1}{q}^{\mathrm{\ell }};q)}_{\mathrm{\ell }}$ written as ${\prod }_{j=0}^{\mathrm{\ell }-1}(1-2a{q}^{j}x+a^2{q}^{2j})$, substitute for $x$ and the parameters according to the left-hand side of the present formula, take the limit, and obtain the ${}_3{}^{}\varphi _2^{}$ as in (18.27.5).

Referenced by:

(18.28.27), (18.28.28), §18.28(x)

Permalink:

http://dlmf.nist.gov/18.28.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

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