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Zentralblatt MATH Database 1931 – 2007 c 2007 European Mathematical Society, FIZ Karlsruhe & Springer-Verlag
1079.33013 Zhang, Zhizheng A note on an identity of Andrews. (English) Electron. J. Comb. 12, No.1, Research paper N3, 3 p., electronic only (2005). [ISSN 1077-8926] http://www.emis.de/journals/EJC/Volume 12/Abstracts/v12i1n3.html In this paper, starting from an identity of G. E. Andrews [Adv. Math. 41, 137-172 (1981; Zbl 0477.33001)] and using the q-exponential operator technique, the author derives , for 0 < |q| < 1, the following identity: ∞ X
∞ X (q/bc, q/ce, acdf ; q)n (q/bd, q/de, acdf ; q)n n d q − c qn 2 2 (ad, df ; q)n+1 (q /bcde; q)n (ac, cf ; q)n+1 (q /bcde; q)n n=0 n=0
=d
(q, qd/c, c/d, abcd, acdf, bcdf, acde, cdef, bcde/q; q)∞ . (ac, ad, cf, df, bc, bd, ce, de, abc2 d2 ef /q; q)∞
Youssef Ben Cheikh (Monastir) Keywords : q-exponential operator Classification : ∗ 33D15 Basic hypergeometric functions of one variable 05A30 q-calculus and related topics Cited in ... 1074.11521 Bhargava, S.; Adiga, Chandrashekar; Somashekara, D.D. Number of representations of an integer as a sum of a square and five times a square. (English) J. Indian Math. Soc., New Ser. 65, No.1-4, 249-251 (1998). [ISSN 0019-5839] Summary: In this note we obtain a formula for the number of representations of an integer n(≥ 1) in the form k 2 + 5m2 . Our results follow from Andrews’ generalization of the well-known Ramanujan’s 1 Ψ1 summation [Adv. Math. 41, 137-172 (1981; Zbl 0477.33001)]. Classification : ∗ 11P05 Waring’s problem and variants 11D85 Representation problems of integers 11P81 Elementary theory of partitions Cited in ...
Zentralblatt MATH Database 1931 – 2007 c 2007 European Mathematical Society, FIZ Karlsruhe & Springer-Verlag
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
D.M.Bressoud Keywords : theta function Classification : ∗ 33E05 Elliptic functions and integrals 11P82 Analytic theory of partitions 11P81 Elementary theory of partitions Cited in ...
1079.33013 Zhang, Zhizheng A note on an identity of Andrews. (English) Electron. J. Comb. 12, No.1, Research paper N3, 3 p., electronic only (2005). [ISSN 1077-8926] http://www.emis.de/journals/EJC/Volume 12/Abstracts/v12i1n3.html In this paper, starting from an identity of G. E. Andrews [Adv. Math. 41, 137-172 (1981; Zbl 0477.33001)] and using the q-exponential operator technique, the author derives , for 0 < |q| < 1, the following identity: ∞ X
∞ X (q/bc, q/ce, acdf ; q)n (q/bd, q/de, acdf ; q)n n d q − c qn 2 2 (ad, df ; q)n+1 (q /bcde; q)n (ac, cf ; q)n+1 (q /bcde; q)n n=0 n=0
=d
(q, qd/c, c/d, abcd, acdf, bcdf, acde, cdef, bcde/q; q)∞ . (ac, ad, cf, df, bc, bd, ce, de, abc2 d2 ef /q; q)∞
Youssef Ben Cheikh (Monastir) Keywords : q-exponential operator Classification : ∗ 33D15 Basic hypergeometric functions of one variable 05A30 q-calculus and related topics Cited in ... 1074.11521 Bhargava, S.; Adiga, Chandrashekar; Somashekara, D.D. Number of representations of an integer as a sum of a square and five times a square. (English) J. Indian Math. Soc., New Ser. 65, No.1-4, 249-251 (1998). [ISSN 0019-5839] Summary: In this note we obtain a formula for the number of representations of an integer n(≥ 1) in the form k 2 + 5m2 . Our results follow from Andrews’ generalization of the well-known Ramanujan’s 1 Ψ1 summation [Adv. Math. 41, 137-172 (1981; Zbl 0477.33001)]. Classification : ∗ 11P05 Waring’s problem and variants 11D85 Representation problems of integers 11P81 Elementary theory of partitions Cited in ...
Zentralblatt MATH Database 1931 – 2007 c 2007 European Mathematical Society, FIZ Karlsruhe & Springer-Verlag
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
D.M.Bressoud Keywords : theta function Classification : ∗ 33E05 Elliptic functions and integrals 11P82 Analytic theory of partitions 11P81 Elementary theory of partitions Cited in ...
