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Zentralblatt MATH Database 1931 – 2007 c 2007 European Mathematical Society, FIZ Karlsruhe & Springer-Verlag
1070.33016 Lovejoy, Jeremy; Ono, Ken Hypergeometric generating functions for values of Dirichlet and other L functions. (English) Proc. Natl. Acad. Sci. USA 100, No.12, 6904-6909 (2003). [ISSN 0027-8424; ISSN 1091-6490] http://dx.doi.org/10.1073/pnas.1131697100 Summary: Although there is vast literature on the values of L functions at nonpositive integers, the recent appearance of some of these values as the coefficients of specializations of knot invariants comes as a surprise. Using work of G. E. Andrews [Adv. Math. 41, 173-185 (1981; Zbl 0477.33002); q-series: their development and application in analysis, combinatories, physics, and computer algebra, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics 66 (Am. Math. Soc, Providence, RI) (1986); Ill. J. Math. 36, No. 2, 251-274 (1992; Zbl 0773.05007)], we revisit this old subject and provide uniform and general results giving such generating functions as specializations of basic hypergeometric functions. For example, we obtain such generating functions for all nontrivial Dirichlet L functions. Classification : ∗ 33D15 Basic hypergeometric functions of one variable 05A17 Partitions of integres (combinatorics) 11M41 Other Dirichlet series and zeta functions Cited in ... 0702.33009 Srivastava, Bhaskar Partial theta function expansions. (English) Tˆ ohoku Math. J., II. Ser. 42, No.1, 119-125 (1990). [ISSN 0040-8735] Among the many identities which George Andrews has discovered and proved from Ramanujan’s “Lost Notebook” are several which expand theta functions in terms of partial products of the same function. As an example, set θ(x; q) = (q 2 ; q 2 )∞ (xq; q 2 )∞ (x−1 q; q 2 )∞ , θN (x; q) = (q 2 ; q 2 )∞ (xq; q 2 )N (x−1 q; q 2 )N . Ramanujan stated and A. E. Andrews has proved [Adv. Math. 41, 137-172 and 173-185 (1981; Zbl 0477.33001 and Zbl 0477.33002)] that 2
3
(∗) θ(x; q ) =
X
q 2n θn (x; q). (q 2 ; q 2 )2n
The author proves these identities in a more general setting. As an example, (*) becomes 2
X q 2n −2n (1 − dq 2n ) −q 3 θ(x; q ) = θn (x; q). (x + x−1 + dq 3 ) (q 2 ; q 2 )2n 3
Zentralblatt MATH Database 1931 – 2007 c 2007 European Mathematical Society, FIZ Karlsruhe & Springer-Verlag
D.M.Bressoud Keywords : theta function Classification : ∗ 33E05 Elliptic functions and integrals 11P82 Analytic theory of partitions 11P81 Elementary theory of partitions Cited in ... 0717.33012 Denis, Remy Y. On certain θ-function expansions. (English) Math. Stud. 56, No.1-4, 114-121 (1989). [ISSN 0025-5742] The author proves transformation formulas for bibasic hypergeometric series of the form X (−aq; q)n (−a−1 ; q)n (α1 ; p)n ...(αr ; p)n xn , (q; q)m )(β1 ; p)n ...(βs ; p)n
n≥0
where m is either 2n or 2n + 1. Special cases of this include lemmas used by G. E. Andrews [Adv. Math. 41, 173-185 (1981; Zbl 0477.33002)]. D.M.Bressoud Keywords : bibasic hypergeometric Classification : ∗ 33E05 Elliptic functions and integrals 33D65 Bibasic functions and multiple bases Cited in ...
1070.33016 Lovejoy, Jeremy; Ono, Ken Hypergeometric generating functions for values of Dirichlet and other L functions. (English) Proc. Natl. Acad. Sci. USA 100, No.12, 6904-6909 (2003). [ISSN 0027-8424; ISSN 1091-6490] http://dx.doi.org/10.1073/pnas.1131697100 Summary: Although there is vast literature on the values of L functions at nonpositive integers, the recent appearance of some of these values as the coefficients of specializations of knot invariants comes as a surprise. Using work of G. E. Andrews [Adv. Math. 41, 173-185 (1981; Zbl 0477.33002); q-series: their development and application in analysis, combinatories, physics, and computer algebra, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics 66 (Am. Math. Soc, Providence, RI) (1986); Ill. J. Math. 36, No. 2, 251-274 (1992; Zbl 0773.05007)], we revisit this old subject and provide uniform and general results giving such generating functions as specializations of basic hypergeometric functions. For example, we obtain such generating functions for all nontrivial Dirichlet L functions. Classification : ∗ 33D15 Basic hypergeometric functions of one variable 05A17 Partitions of integres (combinatorics) 11M41 Other Dirichlet series and zeta functions Cited in ... 0702.33009 Srivastava, Bhaskar Partial theta function expansions. (English) Tˆ ohoku Math. J., II. Ser. 42, No.1, 119-125 (1990). [ISSN 0040-8735] Among the many identities which George Andrews has discovered and proved from Ramanujan’s “Lost Notebook” are several which expand theta functions in terms of partial products of the same function. As an example, set θ(x; q) = (q 2 ; q 2 )∞ (xq; q 2 )∞ (x−1 q; q 2 )∞ , θN (x; q) = (q 2 ; q 2 )∞ (xq; q 2 )N (x−1 q; q 2 )N . Ramanujan stated and A. E. Andrews has proved [Adv. Math. 41, 137-172 and 173-185 (1981; Zbl 0477.33001 and Zbl 0477.33002)] that 2
3
(∗) θ(x; q ) =
X
q 2n θn (x; q). (q 2 ; q 2 )2n
The author proves these identities in a more general setting. As an example, (*) becomes 2
X q 2n −2n (1 − dq 2n ) −q 3 θ(x; q ) = θn (x; q). (x + x−1 + dq 3 ) (q 2 ; q 2 )2n 3
Zentralblatt MATH Database 1931 – 2007 c 2007 European Mathematical Society, FIZ Karlsruhe & Springer-Verlag
D.M.Bressoud Keywords : theta function Classification : ∗ 33E05 Elliptic functions and integrals 11P82 Analytic theory of partitions 11P81 Elementary theory of partitions Cited in ... 0717.33012 Denis, Remy Y. On certain θ-function expansions. (English) Math. Stud. 56, No.1-4, 114-121 (1989). [ISSN 0025-5742] The author proves transformation formulas for bibasic hypergeometric series of the form X (−aq; q)n (−a−1 ; q)n (α1 ; p)n ...(αr ; p)n xn , (q; q)m )(β1 ; p)n ...(βs ; p)n
n≥0
where m is either 2n or 2n + 1. Special cases of this include lemmas used by G. E. Andrews [Adv. Math. 41, 173-185 (1981; Zbl 0477.33002)]. D.M.Bressoud Keywords : bibasic hypergeometric Classification : ∗ 33E05 Elliptic functions and integrals 33D65 Bibasic functions and multiple bases Cited in ...
