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17 q-Hypergeometric and Related FunctionsProperties

Β§17.9 Further Transformations of Ο•rr+1 Functions

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Keywords:
q-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 Ο•rr Functions to Further Transformations of Ο•rr+1 Functions.

Suggested 2016-07-16 by Howard Cohl

See also:
Annotations for Ch.17
Contents
  1. Β§17.9(i) Ο•12β†’Ο•22, Ο•13, or Ο•23
  2. Β§17.9(ii) Ο•23β†’Ο•23
  3. Β§17.9(iii) Further Ο•sr Functions
  4. Β§17.9(iv) Bibasic Series

Β§17.9(i) Ο•12β†’Ο•22, Ο•13, or Ο•23

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Notes:
See Gasper and Rahman (2004, p.Β 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 Β§17.9 and Ch.17

F. H. Jackson’s Transformations

17.9.1 Ο•12⁑(a,bc;q,z) =(z⁒a;q)∞(z;q)βˆžβ’Ο•22⁑(a,c/bc,a⁒z;q,b⁒z),
17.9.2 Ο•12⁑(qβˆ’n,bc;q,z) =(c/b;q)n(c;q)n⁒bn⁒ϕ13⁑(qβˆ’n,b,q/zb⁒q1βˆ’n/c;q,z/c),
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Symbols:
(a;q)n: q-Pochhammer symbol (or q-shifted factorial), Ο•sr+1⁑(a0,…,ar;b1,…,bs;q,z) or Ο•sr+1⁑(a0,…,arb1,…,bs;q,z): basic hypergeometric (or q-hypergeometric) function, q: complex base, n: nonnegative integer and z: 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 q/c in the first row of Ο•13⁑(qβˆ’n,b,q/cb⁒q1βˆ’n/c;q,z/c) was replaced by q/z.

Suggested 2016-08-30 by Xinrong Ma

See also:
Annotations for Β§17.9(i), Β§17.9(i), Β§17.9 and Ch.17
17.9.3 Ο•12⁑(a,bc;q,z) =(a⁒b⁒z/c;q)∞(b⁒z/c;q)βˆžβ’Ο•23⁑(a,c/b,0c,c⁒q/(b⁒z);q,q)+(a,b⁒z,c/b;q)∞(c,z,c/(b⁒z);q)βˆžβ’Ο•23⁑(z,a⁒b⁒z/c,0b⁒z,b⁒z⁒q/c;q,q),
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Symbols:
(a;q)n: q-Pochhammer symbol (or q-shifted factorial), Ο•sr+1⁑(a0,…,ar;b1,…,bs;q,z) or Ο•sr+1⁑(a0,…,arb1,…,bs;q,z): basic hypergeometric (or q-hypergeometric) function, (a1,a2,…,ar;q)n: multiple q-Pochhammer symbol, q: complex base and z: 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 Ο•12 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 Β§17.9(i), Β§17.9(i), Β§17.9 and Ch.17
17.9.3_5 Ο•12⁑(a,bc;q,z) =(c/a,c/b;q)∞(c,c/(a⁒b);q)βˆžβ’Ο•23⁑(a,b,a⁒b⁒z/cq⁒a⁒b/c,0;q,q)+(a,b,a⁒b⁒z/c;q)∞(c,a⁒b/c,z;q)βˆžβ’Ο•23⁑(c/a,c/b,zq⁒c/(a⁒b),0;q,q),
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Symbols:
Ο•sr+1⁑(a0,…,ar;b1,…,bs;q,z) or Ο•sr+1⁑(a0,…,arb1,…,bs;q,z): basic hypergeometric (or q-hypergeometric) function, (a1,a2,…,ar;q)n: multiple q-Pochhammer symbol, q: complex base and z: complex variable
Proof sketch:
Derivable from (17.9.13) after replacing a with d/a and then taking dβ†’0. Then replace a, b, c, e with a⁒b⁒z/c, a, b, c respectively.
Referenced by:
Β§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 Β§17.9(i), Β§17.9(i), Β§17.9 and Ch.17
17.9.4 Ο•12⁑(qβˆ’n,bc;q,z) =(c/b;q)n(c;q)n⁒(b⁒zq)n⁒ϕ23⁑(qβˆ’n,q/z,q1βˆ’n/cb⁒q1βˆ’n/c,0;q,q),
17.9.5 Ο•12⁑(qβˆ’n,bc;q,z) =(c/b;q)n(c;q)n⁒ϕ23⁑(qβˆ’n,b,b⁒z⁒qβˆ’n/cb⁒q1βˆ’n/c,0;q,q).

Β§17.9(ii) Ο•23β†’Ο•23

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

Transformations of Ο•23-Series

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See also:
Annotations for Β§17.9(ii), Β§17.9 and Ch.17
17.9.6 Ο•23⁑(a,b,cd,e;q,d⁒e/(a⁒b⁒c)) =(e/a,d⁒e/(b⁒c);q)∞(e,d⁒e/(a⁒b⁒c);q)βˆžβ’Ο•23⁑(a,d/b,d/cd,d⁒e/(b⁒c);q,e/a),
17.9.7 Ο•23⁑(a,b,cd,e;q,d⁒e/(a⁒b⁒c)) =(b,d⁒e/(a⁒b),d⁒e/(b⁒c);q)∞(d,e,d⁒e/(a⁒b⁒c);q)βˆžβ’Ο•23⁑(d/b,e/b,d⁒e/(a⁒b⁒c)d⁒e/(a⁒b),d⁒e/(b⁒c);q,b),
17.9.8 Ο•23⁑(qβˆ’n,b,cd,e;q,q) =(d⁒e/(b⁒c);q)n(e;q)n⁒(b⁒cd)n⁒ϕ23⁑(qβˆ’n,d/b,d/cd,d⁒e/(b⁒c);q,q),
17.9.9 Ο•23⁑(qβˆ’n,b,cd,e;q,q) =(e/c;q)n(e;q)n⁒cn⁒ϕ23⁑(qβˆ’n,c,d/bd,c⁒q1βˆ’n/e;q,b⁒qe),
17.9.10 Ο•23⁑(qβˆ’n,b,cd,e;q,d⁒e⁒qnb⁒c) =(e/c;q)n(e;q)n⁒ϕ23⁑(qβˆ’n,c,d/bd,c⁒q1βˆ’n/e;q,q).

q-Sheppard Identity

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Keywords:
q-Sheppard identity, q-hypergeometric function
See also:
Annotations for Β§17.9(ii), Β§17.9 and Ch.17
17.9.11 Ο•23⁑(qβˆ’n,b,cd,e;q,q)=(e/c,d/c;q)n(e,d;q)n⁒cn⁒ϕ23⁑(qβˆ’n,c,c⁒b⁒q1βˆ’n/(d⁒e)c⁒q1βˆ’n/e,c⁒q1βˆ’n/d;q,q),
17.9.12 Ο•23⁑(a,b,cd,e;q,d⁒ea⁒b⁒c)=(e/b,e/c,c⁒q/a,q/d;q)∞(e,c⁒q/d,q/a,e/(b⁒c);q)βˆžβ’Ο•23⁑(c,d/a,c⁒q/ec⁒q/a,b⁒c⁒q/e;q,b⁒qd)βˆ’(q/d,e⁒q/d,b,c,d/a,d⁒e/(b⁒c⁒q),b⁒c⁒q2/(d⁒e);q)∞(d/q,e,b⁒q/d,c⁒q/d,q/a,e/(b⁒c),b⁒c⁒q/e;q)βˆžΓ—Ο•23⁑(a⁒q/d,b⁒q/d,c⁒q/dq2/d,e⁒q/d;q,d⁒ea⁒b⁒c),
17.9.13 Ο•23⁑(a,b,cd,e;q,d⁒ea⁒b⁒c)=(e/b,e/c;q)∞(e,e/(b⁒c);q)βˆžβ’Ο•23⁑(d/a,b,cd,b⁒c⁒q/e;q,q)+(d/a,b,c,d⁒e/(b⁒c);q)∞(d,e,b⁒c/e,d⁒e/(a⁒b⁒c);q)βˆžΓ—Ο•23⁑(e/b,e/c,d⁒e/(a⁒b⁒c)d⁒e/(b⁒c),e⁒q/(b⁒c);q,q).

Β§17.9(iii) Further Ο•sr Functions

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Notes:
See Gasper and Rahman (2004, pp.Β 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 Β§17.9 and Ch.17

Sears’ Balanced Ο•34 Transformations

With d⁒e⁒f=a⁒b⁒c⁒q1βˆ’n

17.9.14 Ο•34⁑(qβˆ’n,a,b,cd,e,f;q,q)=(e/a,f/a;q)n(e,f;q)n⁒an⁒ϕ34⁑(qβˆ’n,a,d/b,d/cd,a⁒q1βˆ’n/e,a⁒q1βˆ’n/f;q,q)=(a,e⁒f/(a⁒b),e⁒f/(a⁒c);q)n(e,f,e⁒f/(a⁒b⁒c);q)n⁒ϕ34⁑(qβˆ’n,e/a,f/a,e⁒f/(a⁒b⁒c)e⁒f/(a⁒b),e⁒f/(a⁒c),q1βˆ’n/a;q,q).

Watson’s q-Analog of Whipple’s Theorem

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

With n a nonnegative integer

17.9.15 (a⁒q,a⁒q/(d⁒e);q)n(a⁒q/d,a⁒q/e;q)n⁒ϕ34⁑(a⁒q/(b⁒c),d,e,qβˆ’na⁒q/b,a⁒q/c,d⁒e⁒qβˆ’n/a;q,q)=Ο•78⁑(a,q⁒a12,βˆ’q⁒a12,b,c,d,e,qβˆ’na12,βˆ’a12,a⁒q/b,a⁒q/c,a⁒q/d,a⁒q/e,a⁒qn+1;q,a2⁒q2+nb⁒c⁒d⁒e).

Bailey’s Transformation of Very-Well-Poised Ο•78

17.9.16 Ο•78⁑(a,q⁒a12,βˆ’q⁒a12,b,c,d,e,fa12,βˆ’a12,a⁒q/b,a⁒q/c,a⁒q/d,a⁒q/e,a⁒q/f;q,a2⁒q2b⁒c⁒d⁒e⁒f)=(a⁒q,a⁒q/(d⁒e),a⁒q/(d⁒f),a⁒q/(e⁒f);q)∞(a⁒q/d,a⁒q/e,a⁒q/f,a⁒q/(d⁒e⁒f);q)βˆžβ’Ο•34⁑(a⁒q/(b⁒c),d,e,fa⁒q/b,a⁒q/c,d⁒e⁒f/a;q,q)+(a⁒q,a⁒q/(b⁒c),d,e,f,a2⁒q2/(b⁒d⁒e⁒f),a2⁒q2/(c⁒d⁒e⁒f);q)∞(a⁒q/b,a⁒q/c,a⁒q/d,a⁒q/e,a⁒q/f,a2⁒q2/(b⁒c⁒d⁒e⁒f),d⁒e⁒f/(a⁒q);q)βˆžΓ—Ο•34⁑(a⁒q/(d⁒e),a⁒q/(d⁒f),a⁒q/(e⁒f),a2⁒q2/(b⁒c⁒d⁒e⁒f)a2⁒q2/(b⁒d⁒e⁒f),a2⁒q2/(c⁒d⁒e⁒f),a⁒q2/(d⁒e⁒f);q,q).

Sears–Carlitz Transformation

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

With a=qβˆ’n and n a nonnegative integer,

17.9.17 Ο•23⁑(a,b,ca⁒q/b,a⁒q/c;q,a⁒q⁒zb⁒c)=(a⁒z;q)∞(z;q)βˆžβ’Ο•45⁑(a12,βˆ’a12,(a⁒q)12,βˆ’(a⁒q)12,a⁒q/(b⁒c)a⁒q/b,a⁒q/c,a⁒z,q/z;q,q).

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

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See also:
Annotations for Β§17.9(iii), Β§17.9 and Ch.17
17.9.18 (Ο•34⁑(a,b,a⁒b⁒z,a⁒b/za⁒b⁒q12,βˆ’a⁒b⁒q12,βˆ’a⁒b;q,q))2=Ο•45⁑(a2,b2,a⁒b,a⁒b⁒z,a⁒b/za⁒b⁒q12,βˆ’a⁒b⁒q12,βˆ’a⁒b,a2⁒b2;q,q),

provided that the series expansions of both ϕ’s terminate.

Β§17.9(iv) Bibasic Series

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

Mixed-Base Heine-Type Transformations

17.9.19 βˆ‘n=0∞(a;q2)n⁒(b;q)n(q2;q2)n⁒(c;q)n⁒zn=(b;q)∞⁒(a⁒z;q2)∞(c;q)∞⁒(z;q2)βˆžβ’βˆ‘n=0∞(c/b;q)2⁒n⁒(z;q2)n⁒b2⁒n(q;q)2⁒n⁒(a⁒z;q2)n+(b;q)∞⁒(a⁒z⁒q;q2)∞(c;q)∞⁒(z⁒q;q2)βˆžβ’βˆ‘n=0∞(c/b;q)2⁒n+1⁒(z⁒q;q2)n⁒b2⁒n+1(q;q)2⁒n+1⁒(a⁒z⁒q;q2)n.
17.9.20 βˆ‘n=0∞(a;qk)n⁒(b;q)k⁒n⁒zn(qk;qk)n⁒(c;q)k⁒n=(b;q)∞⁒(a⁒z;qk)∞(c;q)∞⁒(z;qk)βˆžβ’βˆ‘n=0∞(c/b;q)n⁒(z;qk)n⁒bn(q;q)n⁒(a⁒z;qk)n,
k=1,2,3,….
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