Sieved partition functions and -binomial coefficients

Authors:
Frank Garvan and Dennis Stanton

Journal:
Math. Comp. **55** (1990), 299-311

MSC:
Primary 11P68; Secondary 05A19, 05A30, 11B65

MathSciNet review:
1023761

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Abstract: The *q*-binomial coefficient is a polynomial in *q*. Given an integer *t* and a residue class *r* modulo *t*, a sieved *q*-binomial coefficient is the sum of those terms whose exponents are congruent to *r* modulo *t*. In this paper explicit polynomial identities in are given for sieved *q*-binomial coefficients. As a limiting case, generating functions for the sieved partition function are found as multidimensional theta functions. A striking corollary of this representation is the proof of Ramanujan's congruences , and 11 by exhibiting symmetry groups of orders 5, 7, and 11 of explicit quadratic forms. We also verify the Subbarao conjecture for , , and .

**[1]**George E. Andrews,*The theory of partitions*, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. Encyclopedia of Mathematics and its Applications, Vol. 2. MR**0557013****[2]**George E. Andrews and F. G. Garvan,*Dyson’s crank of a partition*, Bull. Amer. Math. Soc. (N.S.)**18**(1988), no. 2, 167–171. MR**929094**, 10.1090/S0273-0979-1988-15637-6**[3]**G. H. Hardy and E. M. Wright,*An introduction to the theory of numbers*, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR**568909****[4]**M. D. Hirschhorn and M. V. Subbarao,*On the parity of*, preprint.**[5]**Oddmund Kolberg,*Some identities involving the partition function*, Math. Scand.**5**(1957), 77–92. MR**0090608****[6]**O. Kolberg,*Note on the parity of the partition function*, Math. Scand.**7**(1959), 377–378. MR**0117213****[7]**M. V. Subbarao,*Some remarks on the partition function*, Amer. Math. Monthly**73**(1966), 851–854. MR**0201409****[8]**Lasse Winquist,*An elementary proof of 𝑝(11𝑚+6)≡0(𝑚𝑜𝑑11)*, J. Combinatorial Theory**6**(1969), 56–59. MR**0236136**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1990-1023761-1

Keywords:
*q*-binomial coefficient,
partitions

Article copyright:
© Copyright 1990
American Mathematical Society