A fundamental modular identity and some applications

Authors:
Richard Blecksmith, John Brillhart and Irving Gerst

Journal:
Math. Comp. **61** (1993), 83-95

MSC:
Primary 11P83; Secondary 05A19, 11F11

MathSciNet review:
1197509

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a six-parameter identity whose terms have the form , where . This identity is then used to give a new proof of the familiar Ramanujan identity , where and . Two other identities, called "balanced identities", are also established through its use.

**[1]**B. J. Birch,*A look back at Ramanujan’s notebooks*, Math. Proc. Cambridge Philos. Soc.**78**(1975), 73–79. MR**0379372****[2]**Richard Blecksmith, John Brillhart, and Irving Gerst,*Parity results for certain partition functions and identities similar to theta function identities*, Math. Comp.**48**(1987), no. 177, 29–38. MR**866096**, 10.1090/S0025-5718-1987-0866096-X**[3]**Richard Blecksmith, John Brillhart, and Irving Gerst,*Some infinite product identities*, Math. Comp.**51**(1988), no. 183, 301–314. MR**942157**, 10.1090/S0025-5718-1988-0942157-2**[4]**Richard Blecksmith, John Brillhart, and Irving Gerst,*On a certain (mod 2) identity and a method of proof by expansion*, Math. Comp.**56**(1991), no. 194, 775–794. MR**1068825**, 10.1090/S0025-5718-1991-1068825-2**[5]**-,*A general formula for balanced**identities of a simple type*, Abstracts Amer. Math. Soc.**13**(1992), 504.**[6]**D. Bressoud,*Proof and generalization of certain identities conjectured by Ramanujan*, PhD Thesis, Temple University, Philadelphia, PA, 1977.**[7]**S. Ramanujan,*Algebraic relations between certain infinite products [Proc. London Math. Soc. (2) 18 (1920), Records for 13 March 1919]*, Collected papers of Srinivasa Ramanujan, AMS Chelsea Publ., Providence, RI, 2000, pp. 231. MR**2280872****[8]**S. Robins,*Arithmetic properties of modular forms*, PhD Thesis, University of California, Los Angeles, CA, 1991.**[9]**L. J. Rogers,*On a type of modular relation*, Proc. London Math. Soc. (2)**19**(1921), 387-397.**[10]**G. N. Watson,*Proof of certain identities in combinatory analysis*, J. Indian Math. Soc.**20**(1933), 57-69.

Retrieve articles in *Mathematics of Computation*
with MSC:
11P83,
05A19,
11F11

Retrieve articles in all journals with MSC: 11P83, 05A19, 11F11

Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1993-1197509-7

Keywords:
Triple and quintuple product,
modular identity,
balanced and identity

Article copyright:
© Copyright 1993
American Mathematical Society