RECURRENCE RELATION FOR COMPUTING A BIPARTITION FUNCTION

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1 ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 48, Number, 08 RECURRENCE RELATION FOR COMPUTING A BIPARTITION FUNCTION D.S. GIREESH AND M.S. MAHADEVA NAIKA ABSTRACT. Recently, Merca [4] found the recurrence relation for computing the partition function p(n which requires only the values of p(k for k n/. In this article, we find the recurrence relation to compute the bipartition function p (n which requires only the values of p (k for k n/. In addition, we also find recurrences for p(n q(n (number of partitions of n into distinct parts, relations connecting p(n q 0 (n (number of partitions of n into distinct odd parts.. Introduction. A partition of a positive integer n is a nonincreasing sequence of positive integers whose sum is n. Let p(n denote the number of partitions of n, p (n denote the number of bipartitions of n, q(n denote the number of partitions of n into distinct parts q o (n denote the number of partitions of n into distinct odd parts. Throughout the paper, we set p(0 p (0 q(0 q o (0 p(x p (x q(x q o (x 0 if x < 0. (. The generating functions for p(n, p (n, q(n q o (n are p(nq n, (q; q (. p (nq n (q; q, 00 AMS Mathematics subject classification. Primary 05A7, P8, P8. Keywords phrases. Recurrence relation, partition, bipartition. Received by the editors on October 9, 06, in revised form on December 9, 06. DOI:0.6/RMJ Copyright c 08 Rocky Mountain Mathematics Consortium 37

2 38 D.S. GIREESH AND M.S. MAHADEVA NAIKA (.3 (.4 q(nq n (q; q ( q; q, q o (nq n ( q; q, where q < (a; q ( a( aq( aq is the q-shifted factorial. Euler [] invented generating function (. which gives rise to a recurrence relation for p(n, ( (.5 ( k k(3k p n δ 0,n, where δ i,j is the Kronecker delta. To compute partition function p(n using (.5 requires the values of p(k with k n. Numerous mathematicians have given other recurrence relations for the partition function p(n. In 004, Ewell [3] found two recurrence relations for p(n: ( n k(k + / (.6 p(n p + ( k p(n k 4 (.7 k ( n k(k + / p(n p + ( k {p(n k(3k + p(n k(3k + }, k which requires the values of p(k with k n to compute p(n. Over the years, it has been a challenge for mathematicians to find the recurrence relation for p(n that requires less number of values of p(k with k < n. In 06, using Ramanujan s theta function, Merca [4] found the most efficient recurrence relation (.8 p(n n/ j ( n p(kp k j(4j + ( n, which requires only the values of p(k with k n/ to compute p(n.

3 BIPARTITION FUNCTION RECURRENCE RELATION 39 Inspired by their relations, in this paper, we find the recurrence relation for bipartition function p (n that requires only the values of p (k with k n/. In addition, we also find recurrences for p(n q(n, the relation connecting p(n q o (n. Ramanujan s theta functions Jacobi s identity play a key role in proving our main results. For q <, Ramanujan s theta functions [, page 36, entry ] are defined as (.9 ψ(q q n(n+/ (.0 f( q (q; q n n q n(n+ (q ; q (q; q ( n q n(3n /. Lemma. (Jacobi s identity. [, page 39, entry 4]. We have (. (q; q 3 ( m (m + q m(m+/. m0 Our main result is stated in the next theorem. Theorem.. For each integer n 0, (. p (n i,j n/ p ( n + i,j p (k k j(4j i(4i + ( n n / p ( n p (k k j(4j 3 i(4i ( n.

4 40 D.S. GIREESH AND M.S. MAHADEVA NAIKA More explicitly, the above result may be written as (.3 n p (n p (kp (n k j(4j i(4i (.4 i,j + p (n + i,j n p (kp (n k j(4j 3 i(4i 3 i,j n p (kp (n k j(4j 3 i(4i. Example.3. We see by Theorem. that the values of p (n for n {0,,, 3, 4, 5, 6, 7} are: p 0, p p 0, p p 0 (p 0 + p 5, p 3 p 0 (p 0 + p 0, p 4 p 0 (p 0 + p + p + p 0, p 5 4p 0 (p + p + p 36, p 6 p o (3p 0 + 4p + p + p 3 + p (p + p 65, p 7 p 0 (p 0 + p + p 3 + p (p + p 0, where here, throughout this example, we set p (n p n. With the above values in h, we can compute the values of p (4 p (5, i.e., p (4 p 0 (p + p + 3p 4 + p 5 + p 6 + p 7 + p (p + 3p 3 + p 4 + p 5 + p 6 + p (3p + 4p 3 + p 4 + p 5 + p 3 (p 3 + p p (5 p 0 (p 0 + p + p + p 3 + p 4 + p 6 + p 7 + p (p + p + p 3 + p 5 + p 6 + p (p + p 4 + p 5 + p 3 (p 3 + p

5 BIPARTITION FUNCTION RECURRENCE RELATION 4. Proof of Theorem.. We write (. (q; q (q; q (q ; q Substituting (.9 into (., we obtain (. (q; q (q ; q (q ; q (q; q. q k(k+. Replacing q by q in equation (., we find that (.3 ( q; q (q ; q ( k q k(k+. Therefore, we can write p(nq n ( (.4 + (q; q ( q; q (q ; q ( + ( k q k(k+ (q ; q q k(4k+ p (nq n q k(4k, which is equivalent to (.5 p(nq n p (nq n q k(4k. Using the Cauchy product of two power series, we find that (.6 p(nq n p (n k(4k q n. Equating coefficients of q n, we obtain (.7 p(n p (n k(4k.

6 4 D.S. GIREESH AND M.S. MAHADEVA NAIKA In a similar fashion, considering (.8 p(n + q n+ ( (q; q ( q; q we derive the following expression of p(n + in terms of p (n: (.9 p(n + Now, we consider (.0 p (n k(4k 3. (q; q (q ; q (q ; q. (q; q (q; q Using (., (., (.9 in (.0, we find that p (nq n p (kq k p(nq n Using the Cauchy product of power series, we have (. p (nq n j j, q j(j+. p (kp(n k j(j + q n. Equating coefficients of q n on both sides of (., we obtain (. p (n j n/ p (kp(n k j(j + j j + j n/ n/ n/ p (kp(n k j(j p (kp(n k j(4j p (kp(n k j(4j 3.

7 BIPARTITION FUNCTION RECURRENCE RELATION 43 Replacing n by n n by n + in (., we find that n (.3 p (n p (kp(n k j(4j j (.4 n + p (kp(n k j(4j 3 j n p (n + p (kp(n + k j(4j j + j n p (kp(n k j(4j 3. Using (.7 (.9 in (.3 (.4, we arrive at (.. 3. New recurrences for p(n q(n. Theorem 3.. For each nonnegative integer n, we have (3. ( k+n (k + p(n k(k + where l m are integers. { ( l+m if n l(3l / + m(3m, 0 otherwise, Theorem 3.. For each integer n 0, we have ( { (3. ( k k(3k ( l if n l(3l, q n 0 otherwise, where l is an integer. Proof of Theorem 3.. We have ( q; q (q ; q 3 (q; q (q 4 ; q 4,

8 44 D.S. GIREESH AND M.S. MAHADEVA NAIKA that is, (q ; q 3 ( q; q (q; q (q 4 ; q 4. Using (.0 (. in the above equation, we obtain (3.3 ( k+n (k + p(nq n+k(k+ n, l,m ( l+m q l(3l /+m(3m. Result (3. follows from (3.3 by extracting like powers of q. Proof of Theorem 3.. We write which is equivalent to (3.4 (q; q (q; q (q ; q, (q; q (q; q (q ; q. Substituting (.3 (.0 into (3.4, we find that (3.5 q(nq n ( k q k(3k / l ( l q l(3l, from which the result (3. follows. 4. Relation connecting p(n q o (n. Theorem 4.. For each n 0, (4. ( n p ( n p k(k 3 + ( n k(k ( n 3 4 ( n q o (n.

9 BIPARTITION FUNCTION RECURRENCE RELATION 45 Proof. Equation (.0 can be expressed as (4. (q; q (q ; q ( k q k(3k /. Replacing q by q in (4., we obtain (4.3 ( q; q (q ; q However, we have ( k(3k+/ q k(3k /. q o (nq n (q; q + ( q; q (q ; q (( k + ( k(3k+/ q k(3k / ( (q ; q q k(k q, k(k+7+ which is equivalent to ( q o (nq n p(nq n q k(k q. k(k+7+6 Using the Cauchy product of two power series, we find that (4.4 q o (nq n p(n k(k q n p(n k(k + 7 6q n. Equating coefficients of q n on both sides of (4.4, we obtain (4.5 q o (n p(n k(k p(n k(k

10 46 D.S. GIREESH AND M.S. MAHADEVA NAIKA By taking q o (n + q n+ ( q; q (q; q, we also find in a similar fashion that (4.6 q o (n+ p(n k(k 5 Combining (4.5 (4.6, we arrive at (4.. Example 4.. If n 3, p( p(8 p(9 + p(4 9, p(n k(k+. q 0 (3 equals 9 since the nine partitions in question are: 3, , , , , , , , It would be interesting to find the recurrence relation for a t- tuple partition function denoted by p t (n, which would lead to a generalization of (.8 (.. Acknowledgments. The authors would like to thank an anonymous referee for helpful comments. The first author would like to thank Prof. N.D. Baruah for information about the article [4] in a GIAN course at Tezpur University. REFERENCES. B.C. Berndt, Ramanujan s notebooks, Part III, Springer-Verlag, New York, 99.. L. Euler, Introduction to analysis of the infinite, Springer-Verlag, New York, J.A. Ewell, Recurrences for the partition function its relatives, Rocky Mountain J. Math. 34 (004, M. Merca, Fast computation of the partition function, J. Num. Th. 64 (06,

11 BIPARTITION FUNCTION RECURRENCE RELATION S. Ramanujan, Collected papers, Cambridge University Press, Cambridge, 97. Bangalore University, Central College Campus, Department of Mathematics, Bengaluru , Karnataka, India M.S. Ramaiah University of Applied Sciences, Department of Mathematics, Peenya Campus, #470-P, Peenya Industrial Area, Peenya 4th Phase, Bengaluru , Karnataka, India address: Bangalore University, Central College Campus, Department of Mathematics, Bengaluru , Karnataka, India address:

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