17 q-Hypergeometric and Related Functions Properties 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions 17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions

§17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions

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Keywords:: -hypergeometric function, transformations
Referenced by:: Erratum (V1.0.12) for Section 17.9
Permalink:: http://dlmf.nist.gov/17.9
Correction (effective with 1.0.12):: The title for this section was changed from Transformations of Higher ${{}_{r}\phi_{r}}$ Functions to Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions.
Suggested 2016-07-16 by Howard Cohl
See also:: Annotations for Ch.17

§17.9(i) ${{}_{2}\phi_{1}}\to{{}_{2}\phi_{2}}$ , ${{}_{3}\phi_{1}}$ , or ${{}_{3}\phi_{2}}$
§17.9(ii) ${{}_{3}\phi_{2}}\to{{}_{3}\phi_{2}}$
§17.9(iii) Further ${{}_{r}\phi_{s}}$ Functions
§17.9(iv) Bibasic Series

§17.9(i) ${{}_{2}\phi_{1}}\to{{}_{2}\phi_{2}}$ , ${{}_{3}\phi_{1}}$ , or ${{}_{3}\phi_{2}}$

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Notes:: See Gasper and Rahman (2004, p. 351).
Permalink:: http://dlmf.nist.gov/17.9.i
Addition (effective with 1.1.0):: Equation (17.9.3_5) was added.
Suggested 2019-10-19 by Slobodan Damjanovic
See also:: Annotations for §17.9 and Ch.17

F. H. Jackson’s Transformations

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Keywords:: F. H. Jackson’s transformations, -hypergeometric function
See also:: Annotations for §17.9(i), §17.9 and Ch.17

17.9.1 ${{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(za;q\right)_{\infty}% }{\left(z;q\right)_{\infty}}{{}_{2}\phi_{2}}\left({a,c/b\atop c,az};q,bz\right),$

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Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or -hypergeometric) function, : complex base and : complex variable
Permalink:: http://dlmf.nist.gov/17.9.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17

17.9.2 ${{}_{2}\phi_{1}}\left({q^{-n},b\atop c};q,z\right)=\frac{\left(c/b;q\right)_{n% }}{\left(c;q\right)_{n}}b^{n}{{}_{3}\phi_{1}}\left({q^{-n},b,q/z\atop bq^{1-n}% /c};q,z/c\right),$

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Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or -hypergeometric) function, : complex base, : nonnegative integer and : complex variable
Referenced by:: Erratum (V1.0.14) for Equation (17.9.2)
Permalink:: http://dlmf.nist.gov/17.9.E2
Encodings:: TeX, pMML, png
Correction (effective with 1.0.14):: The entry in the first row of ${{}_{3}\phi_{1}}\left({q^{-n},b,q/c\atop bq^{1-n}/c};q,z/c\right)$ was replaced by .
Suggested 2016-08-30 by Xinrong Ma
See also:: Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17

17.9.3 ${{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(abz/c;q\right)_{% \infty}}{\left(bz/c;q\right)_{\infty}}{{}_{3}\phi_{2}}\left({a,c/b,0\atop c,cq% /(bz)};q,q\right)+\frac{\left(a,bz,c/b;q\right)_{\infty}}{\left(c,z,c/(bz);q% \right)_{\infty}}{{}_{3}\phi_{2}}\left({z,abz/c,0\atop bz,bzq/c};q,q\right),$

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Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol, : complex base and : complex variable
Referenced by:: Erratum (V1.0.23) for Equation (17.9.3)
Permalink:: http://dlmf.nist.gov/17.9.E3
Encodings:: TeX, pMML, png
Correction (effective with 1.0.23):: 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 ${{}_{2}\phi_{1}}$ 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)).
Suggested 2019-04-26 by Roberto S. Costas-Santos
See also:: Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17

17.9.3_5 ${{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(c/a,c/b;q\right)_{% \infty}}{\left(c,c/(ab);q\right)_{\infty}}{{}_{3}\phi_{2}}\left({a,b,abz/c% \atop qab/c,0};q,q\right)+\frac{\left(a,b,abz/c;q\right)_{\infty}}{\left(c,ab/% c,z;q\right)_{\infty}}{{}_{3}\phi_{2}}\left({c/a,c/b,z\atop qc/(ab),0};q,q% \right),$

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Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol, : complex base and : complex variable
Proof sketch:: Derivable from (17.9.13) after replacing with and then taking $d\rightarrow 0$ . Then replace , , , with , , , respectively.
Referenced by:: §17.9(i), Erratum (V1.1.0) for Additions
Permalink:: http://dlmf.nist.gov/17.9.E3_5
Encodings:: TeX, pMML, png
Addition (effective with 1.1.0):: This equation was added.
See also:: Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17

17.9.4 ${{}_{2}\phi_{1}}\left({q^{-n},b\atop c};q,z\right)=\frac{\left(c/b;q\right)_{n% }}{\left(c;q\right)_{n}}\left(\frac{bz}{q}\right)^{n}{{}_{3}\phi_{2}}\left({q^% {-n},q/z,q^{1-n}/c\atop bq^{1-n}/c,0};q,q\right),$

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Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or -hypergeometric) function, : complex base, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/17.9.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17

17.9.5 ${{}_{2}\phi_{1}}\left({q^{-n},b\atop c};q,z\right)=\frac{\left(c/b;q\right)_{n% }}{\left(c;q\right)_{n}}{{}_{3}\phi_{2}}\left({q^{-n},b,bzq^{-n}/c\atop bq^{1-% n}/c,0};q,q\right).$

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Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or -hypergeometric) function, : complex base, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/17.9.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17

§17.9(ii) ${{}_{3}\phi_{2}}\to{{}_{3}\phi_{2}}$

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Notes:: See Gasper and Rahman (2004, pp. 71–73). For (17.9.11) apply (17.9.10) to (17.9.9) and interchange the roles of and .
Permalink:: http://dlmf.nist.gov/17.9.ii
See also:: Annotations for §17.9 and Ch.17

Transformations of ${{}_{3}\phi_{2}}$ -Series

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See also:: Annotations for §17.9(ii), §17.9 and Ch.17

17.9.6 ${{}_{3}\phi_{2}}\left({a,b,c\atop d,e};q,de/(abc)\right)=\frac{\left(e/a,de/(% bc);q\right)_{\infty}}{\left(e,de/(abc);q\right)_{\infty}}{{}_{3}\phi_{2}}% \left({a,d/b,d/c\atop d,de/(bc)};q,e/a\right),$

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Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol and : complex base
Permalink:: http://dlmf.nist.gov/17.9.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

17.9.7 ${{}_{3}\phi_{2}}\left({a,b,c\atop d,e};q,de/(abc)\right)=\frac{\left(b,de/(ab)% ,de/(bc);q\right)_{\infty}}{\left(d,e,de/(abc);q\right)_{\infty}}\*{{}_{3}\phi% _{2}}\left({d/b,e/b,de/(abc)\atop de/(ab),de/(bc)};q,b\right),$

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Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol and : complex base
Permalink:: http://dlmf.nist.gov/17.9.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

17.9.8 ${{}_{3}\phi_{2}}\left({q^{-n},b,c\atop d,e};q,q\right)=\frac{\left(de/(bc);q% \right)_{n}}{\left(e;q\right)_{n}}\left(\frac{bc}{d}\right)^{n}{{}_{3}\phi_{2}% }\left({q^{-n},d/b,d/c\atop d,de/(bc)};q,q\right),$

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Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or -hypergeometric) function, : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.9.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

17.9.9 ${{}_{3}\phi_{2}}\left({q^{-n},b,c\atop d,e};q,q\right)=\frac{\left(e/c;q\right% )_{n}}{\left(e;q\right)_{n}}c^{n}{{}_{3}\phi_{2}}\left({q^{-n},c,d/b\atop d,cq% ^{1-n}/e};q,\frac{bq}{e}\right),$

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Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or -hypergeometric) function, : complex base and : nonnegative integer
Referenced by:: §17.9(ii)
Permalink:: http://dlmf.nist.gov/17.9.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

17.9.10 ${{}_{3}\phi_{2}}\left({q^{-n},b,c\atop d,e};q,\frac{deq^{n}}{bc}\right)=\frac{% \left(e/c;q\right)_{n}}{\left(e;q\right)_{n}}{{}_{3}\phi_{2}}\left({q^{-n},c,d% /b\atop d,cq^{1-n}/e};q,q\right).$

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Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or -hypergeometric) function, : complex base and : nonnegative integer
Referenced by:: §17.9(ii)
Permalink:: http://dlmf.nist.gov/17.9.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

-Sheppard Identity

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Keywords:: -Sheppard identity, -hypergeometric function
See also:: Annotations for §17.9(ii), §17.9 and Ch.17

17.9.11 ${{}_{3}\phi_{2}}\left({q^{-n},b,c\atop d,e};q,q\right)=\frac{\left(e/c,d/c;q% \right)_{n}}{\left(e,d;q\right)_{n}}c^{n}{{}_{3}\phi_{2}}\left({q^{-n},c,% \ifrac{cbq^{1-n}}{(de)}\atop\ifrac{cq^{1-n}}{e},\ifrac{cq^{1-n}}{d}};q,q\right),$

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Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol, : complex base and : nonnegative integer
Referenced by:: §17.9(ii)
Permalink:: http://dlmf.nist.gov/17.9.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

17.9.12 ${{}_{3}\phi_{2}}\left({a,b,c\atop d,e};q,\frac{de}{abc}\right)=\frac{\left(e/b% ,e/c,cq/a,q/d;q\right)_{\infty}}{\left(e,cq/d,q/a,e/(bc);q\right)_{\infty}}{{}% _{3}\phi_{2}}\left({c,d/a,cq/e\atop cq/a,bcq/e};q,\frac{bq}{d}\right)-\frac{% \left(q/d,eq/d,b,c,d/a,de/(bcq),bcq^{2}/(de);q\right)_{\infty}}{\left(d/q,e,bq% /d,cq/d,q/a,e/(bc),bcq/e;q\right)_{\infty}}{{}_{3}\phi_{2}}\left({aq/d,bq/d,cq% /d\atop q^{2}/d,eq/d};q,\frac{de}{abc}\right),$

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Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol and : complex base
Permalink:: http://dlmf.nist.gov/17.9.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

17.9.13 ${{}_{3}\phi_{2}}\left({a,b,c\atop d,e};q,\frac{de}{abc}\right)=\frac{\left(e/b% ,e/c;q\right)_{\infty}}{\left(e,e/(bc);q\right)_{\infty}}{{}_{3}\phi_{2}}\left% ({d/a,b,c\atop d,bcq/e};q,q\right)+\frac{\left(d/a,b,c,de/(bc);q\right)_{% \infty}}{\left(d,e,bc/e,de/(abc);q\right)_{\infty}}{{}_{3}\phi_{2}}\left({e/b,% e/c,de/(abc)\atop de/(bc),eq/(bc)};q,q\right).$

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Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol and : complex base
Referenced by:: (17.9.3_5)
Permalink:: http://dlmf.nist.gov/17.9.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

§17.9(iii) Further ${{}_{r}\phi_{s}}$ Functions

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Notes:: See Gasper and Rahman (2004, pp. 43, 50, 63–64, 70, 72, 361).
Referenced by:: Erratum (V1.1.12) for Subsection 17.9(iii)
Permalink:: http://dlmf.nist.gov/17.9.iii
See also:: Annotations for §17.9 and Ch.17

Sears’ Balanced ${{}_{4}\phi_{3}}$ Transformations

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Keywords:: Sears’ balanced ${{}_{4}\phi_{3}}$ transformation, -hypergeometric function
See also:: Annotations for §17.9(iii), §17.9 and Ch.17

With $def=abcq^{1-n}$

17.9.14 ${{}_{4}\phi_{3}}\left({q^{-n},a,b,c\atop d,e,f};q,q\right)=\frac{\left(e/a,f/a% ;q\right)_{n}}{\left(e,f;q\right)_{n}}a^{n}{{}_{4}\phi_{3}}\left({q^{-n},a,d/b% ,d/c\atop d,aq^{1-n}/e,aq^{1-n}/f};q,q\right)=\frac{\left(a,ef/(ab),ef/(ac);q% \right)_{n}}{\left(e,f,ef/(abc);q\right)_{n}}{{}_{4}\phi_{3}}\left({q^{-n},e/a% ,f/a,ef/(abc)\atop ef/(ab),ef/(ac),q^{1-n}/a};q,q\right).$

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Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol, : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.9.E14
Encodings:: TeX, pMML, png
See also:: Annotations for §17.9(iii), §17.9(iii), §17.9 and Ch.17

Watson’s -Analog of Whipple’s Theorem

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Keywords:: Watson’s -analog, Whipple’s theorem
See also:: Annotations for §17.9(iii), §17.9 and Ch.17

With a nonnegative integer

17.9.15 $\frac{\left(aq,aq/(de);q\right)_{n}}{\left(aq/d,aq/e;q\right)_{n}}{{}_{4}\phi_% {3}}\left({aq/(bc),d,e,q^{-n}\atop aq/b,aq/c,deq^{-n}/a};q,q\right)={{}_{8}% \phi_{7}}\left({a,qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,d,e,q^{-n}\atop a^{% \frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e,aq^{n+1}};q,\frac{a^{2}q^{2+% n}}{bcde}\right).$

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Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol, : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.9.E15
Encodings:: TeX, pMML, png
See also:: Annotations for §17.9(iii), §17.9(iii), §17.9 and Ch.17

Bailey’s Transformation of Very-Well-Poised ${{}_{8}\phi_{7}}$

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Keywords:: Bailey’s transformation of very-well-poised ${{}_{8}\phi_{7}}$ , -hypergeometric function
See also:: Annotations for §17.9(iii), §17.9 and Ch.17

17.9.16 ${{}_{8}\phi_{7}}\left({a,qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,d,e,f\atop a^{% \frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e,aq/f};q,\frac{a^{2}q^{2}}{% bcdef}\right)=\frac{\left(aq,aq/(de),aq/(df),aq/(ef);q\right)_{\infty}}{\left(% aq/d,aq/e,aq/f,aq/(def);q\right)_{\infty}}{{}_{4}\phi_{3}}\left({aq/(bc),d,e,f% \atop aq/b,aq/c,def/a};q,q\right)+\frac{\left(aq,aq/(bc),d,e,f,a^{2}q^{2}/(% bdef),a^{2}q^{2}/(cdef);q\right)_{\infty}}{\left(aq/b,aq/c,aq/d,aq/e,aq/f,a^{2% }q^{2}/(bcdef),def/(aq);q\right)_{\infty}}\*{{}_{4}\phi_{3}}\left({aq/(de),aq/% (df),aq/(ef),a^{2}q^{2}/(bcdef)\atop a^{2}q^{2}/(bdef),a^{2}q^{2}/(cdef),aq^{2% }/(def)};q,q\right).$

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Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol and : complex base
Permalink:: http://dlmf.nist.gov/17.9.E16
Encodings:: TeX, pMML, png
See also:: Annotations for §17.9(iii), §17.9(iii), §17.9 and Ch.17

Sears–Carlitz Transformation

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See also:: Annotations for §17.9(iii), §17.9 and Ch.17

With $a=q^{-n}$ and a nonnegative integer,

17.9.17 ${{}_{3}\phi_{2}}\left({a,b,c\atop aq/b,aq/c};q,\frac{aqz}{bc}\right)=\frac{% \left(az;q\right)_{\infty}}{\left(z;q\right)_{\infty}}\*{{}_{5}\phi_{4}}\left(% {a^{\frac{1}{2}},-a^{\frac{1}{2}},(aq)^{\frac{1}{2}},-(aq)^{\frac{1}{2}},aq/(% bc)\atop aq/b,aq/c,az,q/z};q,q\right).$

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Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or -hypergeometric) function, : complex base and : complex variable
Permalink:: http://dlmf.nist.gov/17.9.E17
Encodings:: TeX, pMML, png
See also:: Annotations for §17.9(iii), §17.9(iii), §17.9 and Ch.17

Gasper’s -Analog of Clausen’s Formula (16.12.2)

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See also:: Annotations for §17.9(iii), §17.9 and Ch.17

17.9.18 $\left({{}_{4}\phi_{3}}\left({a,b,abz,ab/z\atop abq^{\frac{1}{2}},-abq^{\frac{1% }{2}},-ab};q,q\right)\right)^{2}={{}_{5}\phi_{4}}\left({a^{2},b^{2},ab,abz,ab/% z\atop abq^{\frac{1}{2}},-abq^{\frac{1}{2}},-ab,a^{2}b^{2}};q,q\right),$

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Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or -hypergeometric) function, : complex base and : complex variable
Permalink:: http://dlmf.nist.gov/17.9.E18
Encodings:: TeX, pMML, png
See also:: Annotations for §17.9(iii), §17.9(iii), §17.9 and Ch.17

provided that the series expansions of both $\phi$ ’s terminate.

§17.9(iv) Bibasic Series

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Keywords:: bibasic sums and series, -hypergeometric function
Notes:: See Andrews (1966b).
Permalink:: http://dlmf.nist.gov/17.9.iv
See also:: Annotations for §17.9 and Ch.17

Mixed-Base Heine-Type Transformations

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Keywords:: mixed base Heine-type transformations, -hypergeometric function
See also:: Annotations for §17.9(iv), §17.9 and Ch.17

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

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Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -shifted factorial), : complex base, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/17.9.E19
Encodings:: TeX, pMML, png
See also:: Annotations for §17.9(iv), §17.9(iv), §17.9 and Ch.17

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

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