Rademacher-Type Formulas for Restricted Partition and Overpartition Functions

Transcription

1 Georgia Southern University Digital Southern Mathematical Sciences Faculty Publications Mathematical Sciences, Department of Rademacher-Type Formulas for Restricted Partition and Overpartition Functions Andrew Sills Georgia Southern University, Follow this and additional wors at: Part of the Mathematics Commons Recommended Citation Sills, Andrew "Rademacher-Type Formulas for Restricted Partition and Overpartition Functions." Ramanujan Journal, 23-3: doi: 0.007/s y source: This article is brought to you for free and open access by the Mathematical Sciences, Department of at Digital Southern. It has been accepted for inclusion in Mathematical Sciences Faculty Publications by an authorized administrator of Digital Southern. For more information, please contact

2 Ramanujan Journal manuscript No. will be inserted by the editor Rademacher-type formulas for restricted partition and overpartition functions Andrew V. Sills Received: date / Accepted: date Dedicated to George Andrews on the occasion of his seventieth birthday Abstract A collection of Hardy-Ramanujan-Rademacher type formulas for restricted partition and overpartition functions is presented, framed by several biographical anecdotes. Keywords partitions circle method Rogers-Ramanujan identities Mathematics Subject Classification 2000 P82 P85 05A9 Introduction When George Andrews matriculated in the Ph.D. program at the University of Pennsylvania in the fall of 96, his intention was to specialize in geometric number theory. He had been attracted to Penn s graduate program in part because the academic year had been designated a special year in number theory there. The academic year culminated in a celebration of the seventieth birthday of Professor Hans Rademacher. Rademacher taught Andrews in his analytic number theory class that year, and there Andrews was introduced to the theory of partitions. A partition λ of an integer n is a wealy decreasing finite sequence of positive integers λ,λ 2,...,λ s whose sum is n. Each λ i is called a part of the partition λ. The theory of integer partitions began with Euler [22], who introduced generating functions to study pn, the number of partitions of n, and found that the generating function for pn was representable as an elegant infinite product: pn n m m.. The circle method was created by Hardy and Ramanujan and later improved by Rademacher, in connection with the study of the function pn, the number of partitions of the integer n. Andrew V. Sills Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA , USA Tel.: Fa: ASills@GeorgiaSouthern.edu

3 2 The circle method has proved to be one of the most useful tools in the history of analytic number theory. Epositions of the circle method may be found in [2,9,53,54,56]. Rademacher s formula for pn is given by 2 pn h,e 2inh/ d sinh 3 n 24 2 dn n 24,.2 h, where h, is a 24th root of unity that frequently occurs in the study of modular forms and is given by h, { h { h ep i 4 2 h h+ 2 2h H + h2 H } {, if 2 h, ep i h H + h2 H }, if 2, a b is the Legendre-Jacobi symbol, and H is any solution of the congruence hh mod. Andrews reports [66] that the formula for pn... was a revolutionary and surprising achievement. The form of this formula is even more stunning. It involves transcendental numbers and epressions that seem to be totally unrelated that might be appropriate, say, in a course on engineering or theoretical physics, but for actually counting how many ways you can add up sums to get a particular number, they seem absolutely incredible. In fact, I was stunned the first time I saw this formula. I could not believe it, and the eperience of seeing it eplained and understanding how it too shape really, I thin, convinced me that this was the area of mathematics that I wanted to pursue. Many practitioners, including a number of Ph.D. students and postdocs who wored under Rademacher, have used the circle method to study various restricted partition functions, often associated with sets of partitions enumerated in famous theorems. These practitioners included Grosswald [24, 25], Haberzetle [26], Hagis [27 35], Hua [39], Isei [40 42], Lehner [43], Livingood [44], Niven [52], and Subramanyasastri [65]. Let us consider several eamples. Theorem Euler, 748 Let qn denote the number of partitions of n into odd parts. Let rn denote the number of partitions of n into distinct parts. Then qn rn for all integers n. Theorem 2 Hagis, 963 qn 24n+ 2 h, 2nh/ h, e 2h, I 24n+ 6 2,.3 where I ν z : r0 2 z ν+2r r!γν + r + is the Bessel function of purely imaginary argument..4

4 3 Theorem 3 Schur, 926 Let sn denote the number of partitions of n into parts congruent to ± mod 6. Let tn denote the number of partitions λ of n where λ i λ i+ 3 and λ i λ i+ > 3 if 3 λ i. Then sn tn for all n. Theorem 4 Niven, 940 and sn 36n 3 d 2d 3 d 6,6d h, e 2nh/ Recently, the author found [6] pn 2 2 h,6h/d,/d 2h d,2, 3h d,2 d,3, d,3 I d2n h, podn 2 d 25d 7 6 d 4,4d h, h, 2 d 2h, e 2inh/ sinh dn n h, 4h/d,/d e 2inh/ d 2h d,2, dn d,2 sinh n d8n 4 8n.6,.7 where podn denotes the number of partitions of n where no odd part is repeated, and pn denotes the number of overpartitions of n. An overpartition of n is a finite wealy decreasing sequence of positive integers where the last occurrence of a given part may or may not be overlined. Thus the eight overpartitions of 3 are 3, 3, 2,, 2,, 2,, 2,,,,,,,. Overpartitions were introduced by S. Corteel and J. Lovejoy in [9] and have been studied etensively by them and others including Bringmann, Chen, Fu, Goh, Hirschhorn, Hitczeno, Lascou, Mahlburg, Robbins, Rødseth, Sellers, Yee, and Zho [, 6 2, 23, 37, 38, 45 5, 57, 58]. Recently, Bringmann and Ono [2] have given eact formulas for the coefficients of all harmonic Maass forms of weight 2. All of the generating functions considered herein are wealy holomorphic modular forms of weight either 0 or 2, and thus they are harmonic Maass forms of weight 2. Accordingly, all of the eact formulas for restricted partition and overpartition functions presented here could be derived from the general theorem in [2]. In this article, we will present several anecdotes from the professional life of George Andrews, and present some new Rademacher type formulas related to the events described.

5 4 2 Identities in the Lost Noteboo Certainly one of the most eciting incidents of George Andrews professional life was his unearthing of Ramanujan s lost noteboo at the Wren Library at Trinity College, Cambridge University in 976 see [4, pp. 5 6,.5] and [6, p. ff] for a full account. As is now well nown, the lost noteboo contains many identities of the Rogers-Ramanujan type. Many of the infinite products appearing in these identities are easily identified as generating functions for certain restricted classes of partitions or overpartitions. The methods of Rademacher may be applied to find eplicit formulas for the coefficients appearing in the series epansions of these generating functions. Below is a list of some of the Rogers-Ramanujan type identities which appear in the lost noteboo. Some of these identities also appear in Slater [59]. Specifically, Eq. 2. is Slater s 6; Eq. 2.3 is Slater s 2; Eq. 2.5 is Slater s 22; Eq. 2.6 is Slater s 25; Eq. 2.7 is Slater s 28; Eq. 2.8 is Slater s 29; and Eq. 2. is Slater s 50. The standard abbreviations a;b j j i0 ab i, a j : a; j will be used. Note that a;b 0. Here and throughout, we assume < to guarantee convergence. n2 n n ; 2 n n2 n n ; 2 n+ nn+/2 n n n2 2 ; 2 n 2n+ + 3m 2 + 3m 3m 2 3m + 3m 2 + 3m 3m 2 3m + 2m [7, Ent 4.2.8] 2. [7, Ent 4.2.9] 2.2 2m [7, Ent.7.4] m 2m [7, Ent.7.3] 2.4 nn+ n ; 2 n+ n n2 ; 2 n 4 ; 4 n nn+ 2 ; 2 n 2n+ 6m 6m 6m 5 m 2m 3m 2m 6m 5 6m 4m [7, Ent 4.2.2] 2.5 [7, Ent 4.2.7] 2.6 2m 2m 9 2m 3 m [7, Ent 4.3.2] 2.7

6 5 n2 ;q 2 n 2n nn+/2 2 ; 2 n n ; 2 n+ nn+/2 ; 2 n n ; 2 n nn+2 ; 2 n 2n+ 6m 2m 6 m [7, Ent ] m 2m 8m 4 [7, Ent..7.5] 2.9 4m 8m 4 + m m [7, Ent..7.4] 2m 2m 0 2m 9 m [7, Ent ] Let us denote the cöefficient of n in the power series epansion of equation j above by R j n. The following combinatorial interpretations are then immediate: R 2. n R 2.2 n the number of overpartitions of n into nonmultiples of 3. R 2.3 n R 2.4 n the number of overpartitions of n with only odd parts. R 2.5 n the number of overpartitions of n where nonoverlined parts are congruent to ±2,3 mod 6. R 2.7 n the number of partitions of n into parts not congruent to 0,±3 mod 2. R 2.9 n the number of overpartitions of n where the nonoverlined parts are odd or congruent to 4 mod 8 R 2. n the number of partitions of n into parts not congruent to 0,±2 mod 2. The circle method yields the following formulas, which are believed to be new to the literature. It could be argued that a number of them capture much of the elegance of the formula for pn. They were found with the aid of Mathematica program written by the author. For a discussion of the automation of certain ey steps of the circle method, along with additional eamples of Rademacher type formulas for restricted partition and overpartition functions, please see [62]. As noted earlier, they could also be derived using the results of Bringmann and Ono [2]. R 2. n 3 2n 2,3 h, e 2inh/ h,2 6h, 2h,3h, 2 I 2n R 2.3 n 4 n 2 h, e 2inh/ h,2 4h, 2h, 3 I n R 2.5 n 2 8n+6 2,6 e h, 3h h, 2inh/,3,,3 6h,6,,6 I 6n

7 6 R 2.6 n 3 d d n 33 d {,4,2} e h, h, 4h 2inh/,2d d,4, d,4 3h d,3, d,3 I 6h d,6, d,6 2 2h d,2, d,2 6d d 2 28n R 2.7 n 4 90n+30 d 39d 2 52d + 28 d 6 h, e 2ihn/ h, 6h 3h d,3, d,3 d,6, d,6 2h d,2, d,2,2d 8+8d d I 2 3n R 2.8 n 3 2d 2 + d n d {,4,2} h, R 2.9 n 3 4 8n+2 2,2d e 2ihn/ h, 2h d, d 84+6d d 2 I 2 24n 6h d,6, 2 33 d,6 R 2.0 n 3 8 n 2 h, h, e 2ihn/ h,2 4h, 2h, 2 8h, I e 2ihn/ h,2 8h, 4h, 2 2h, I 2n+3 4 3n R 2. n 6 24n+5 h, 2ihn/ e h, 4 d 2 d7d 2 46d h d,4, d,4 2h d,2, d,2 6h 2h d, d 2 I d,6, d,6,2d 8n+5d 2 4d

8 7 3 Capparelli s Conjecture The year 992 mared the one hundredth anniversary of the birth of Rademacher, and on July 2 25 of that year a conference honoring the memory of Rademacher was held at Penn State, and George Andrews was of course one of the conference organizers. On the first day of the conference, James Lepowsy of Rutgers gave a tal in which he mentioned that his student Stefano Capparelli had conjectured the following partition identity [3] as a result of his studies of the standard level 3 modules associated with the Lie algebra A 2 2 : Theorem 5 Capparelli Let Cn denote the number of partitions of n into parts ±2, ±3 mod 2. Let Dn denote the number of partitions λ λ,λ 2,... of n such that λ j λ j+ 2, λ j λ j+ 2 only if λ j mod 3, and λ j λ j+ 3 only if λ j is a multiple of 3. Then Cn Dn for all n. This identity is clearly similar in the spirit of those in the classical literature such as Schur s identity our Theorem 3, yet was new. Needless to say, Andrews and others at the conference were quite intrigued by the conjecture. Andrews wored intently for the net several evenings, and was able to find a proof [5] of the identity in time to present it as his tal on the last day of the conference. Of this proof, Andrews wrote [, p. 505], In my proof of Capparelli s conjecture, I was completely guided by the Wilf-Zeilberger method, even if I didn t use Doron s program eplicitly. I couldn t have produced my proof without nowing the principle behind WZ. Although Andrews WZ-inspired proof see [64, 67 70] was the first proof of the Capparelli conjecture, Lie theoretic proofs were later found by Tamba and Xie [63] and Capparelli himself [4]. The generating function for the partitions enumerated by the Cn in Capparelli s identity is Cn n 2m 0 2m 9 2m 3 2m 2, m and indeed the Rademacher method may be applied to find an eplicit formula for Cn. Cn 24n d {,2,3,2} 2h 2nh/ d e d h, I 2+308d + 2d 2 2d 3,2d 3h d,3, 2h d,3 d,2, d,2 6h d,6, d,6 2 4h d,4, d,4 24n 20 23d + 9d 2 6d

9 8 4 The Bailey Chain Of course, Andrews has contributed a large number of important and useful discoveries to the body of mathematical nowledge. One of this author s favorites is the Bailey chain, i.e. the realization that the Bailey lemma is self-replicating and therefore any Bailey pair implies infinitely many others. In particular, every Rogers-Ramanujan type identity is automatically part of an infinite family see [3,4]. The Bailey chain provides an eplanation and a contet for many infinite family q-series identities and their combinatorial counterparts. For eample, David Bressoud s identity [0, p. 5, Eq. 3.4 with r] n,n 2,...,n r 0 N2 +N N2 r n n2 nr 2 2 ; 2 nr 2rm r rm m, 4. where N j : n j + n j+ + +n r and r 2, follows from inserting the Bailey pair α n a, n n2 a 2n a 2 ; 2 n a 2 ; 2 n, β n a, 2 ; 2 n into a certain limiting case of the Bailey chain [4, p. 30, Theorem 3.5], setting a, and then applying Jacobi s triple product identity [4, p. 63, Eq. 7.]. Although Bressoud s combinatorial counterpart to [0, p. 5, Eq. 3.4] ecludes the special case with r our 4. above, the author provided a combinatorial interpretation [60, p. 35, Theorem 6.9], which we recall here: Theorem 6 For r 2, let B r n denote the number of partitions λ λ,λ 2,... of n such that appears as a part less than r times, λ j λ j+r 2, and if λ j λ j+r 2, then r 2 h0 λ j+h r mod 2. For r 3, let A r n denote the number of partitions of n such that no part is a multiple of r, for any nonnegative integer j, either r j+ or r j+, but not both, may appear as parts, and let A 2 n denote the number of partitions of n into distinct odd parts. Then A r n B r n for all integers n. Remar The combinatorial interpretation of the A r n was facilitated by ideas advanced by Andrews and Lewis [8]. We conclude with a Rademacher-type formula for the A r n: A r n 2 2 d,r 24n χ 2r + d 2 > 4d,r 2 d r dr h,d 2inh/ h,2rh/d,/d e I 24n 2r + d 2 4d,r 2, 4.2 rh d,r, 6 2r d,r h,

10 9 where { if P is true, and χp 0 if P is false. Acnowledgement The author thans his thesis advisor, George Andrews, for all his help, encouragement, and indness over many years. The author also thans the anonymous referee for helping him to correct a potentially misleading statement in the original version. References. 998 Steele Prizes, Notices Amer. Math Soc G.E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and its Applications, vol. 2, Addison-Wesley, Reading, MA, 976. Reissued, Cambridge, G.E. Andrews, Multiple series Rogers-Ramanujan type identiites, Pacific J. Math G.E. Andrews, q-series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra, Regional conference series in mathematics, no. 66, Providence: American Mathematical Society, G.E. Andrews, Schur s Theorem, Capparelli s conjecture and q-trinomial coeffcients, in Proc. Rademacher Centenary Conf., 992 pp. 4 54, Contemp. Math. 66, Amer. Math. Soc., Providence, G.E. Andrews and B.C. Berdnt, Ramanujan s Lost Noteboo Part I, Springer, G.E. Andrews and B.C. Berndt, Ramanujan s Lost Noteboo Part II, Springer, G.E. Andrews and R.P. Lewis, An algebraic identity of F.H. Jacson and its implicaiton for partitions, Discrete Math T.M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Graduate Tets in Mathematics, vol. 4, 2nd ed., Springer-Verlag, D.M. Bressoud, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer. Math. Soc , no. 227, 54 pp.. K. Bringmann, J. Lovejoy, Dyson s ran, overpartitions, and wea Maass forms, Int. Math. Res. Not. IMRN 2007, no. 9, 34 pp. 2. K. Bringmann and K. Ono, Coefficients of harmonic Maass forms, Proceedings of the 2008 University of Florida Conference on Partitions, q-series, and Modular Forms, Developments in Mathematics series, Springer, to appear. 3. S. Capparelli, Verte operator relations for ane Lie algebras and combinatorial identities, Ph.D. thesis, Rutgers, S. Capparelli, A construction of the level 3 modules for the ane algebra A 2 2 and a new combinatorial identity of the Rogers-Ramanujan type, Trans. Amer. Math. Soc., , no. 2, O-Y. Chan, Some asymptotics for crans, Acta Arith W.Y.C. Chen, J.J.Y. Zhao, The Gaussian coefficients and overpartitions, Discrete Math S. Corteel, W.M.Y. Goh, P. Hitczeno, A local limit theorem in the theory of overpartitions, Algorithmica S. Corteel, P. Hitczeno, Multiplicity and number of parts in overpartitions, Ann. Comb S. Corteel, J. Lovejoy, Overpartitions, Trans. Amer. Math. Soc S. Corteel, J. Lovejoy, A. J. Yee, Overpartitions and generating functions for generalized Frobenius partitions. Mathematics and computer science. III, 5 24, Trends Math., Birhuser, Basel, S. Corteel and O. Mallet, Overpartitions, lattice paths, and Rogers-Ramanujan identities, J. Combin. Theory Ser. A L. Euler, Introductio in Analysin Infinatorum, Marcum-Michaelem Bousquet, Lausanne, A.M. Fu, A. Lascou, q-identities related to overpartitions and divisor functions, Electron. J. Combin #R38, 7 pp. 24. E. Grosswald, Some theorems concerning partitions, Trans. Amer. Math. Soc E. Grosswald, Partitions into prime powers, Mich. Math. J

11 0 26. M. Haberzetle, On some partition functions, Amer. J. Math P. Hagis, A problem on partitions with a prime modulus p 3, Trans. Amer. Math. Soc P. Hagis, Partitions into odd summands, Amer. J. Math P. Hagis, On a class of partitions with distinct summands, Trans. Amer. Math. Soc P. Hagis, Partitions into odd and unequal parts, Amer. J. Math P. Hagis, Partitions with odd summandssome comments and corrections, Amer. J. Math P. Hagis, A correction of some theorems on partitions, Trans. Amer. Math. Soc P. Hagis, On partitions of an integer into distinct odd summands, Amer. J. Math P. Hagis, Some theorems concerning partitions into odd summands, Amer. J. Math P. Hagis, Partitions with a restriction on the multiplicity of summands, Trans. Amer. Math. Soc G.H. Hardy, S. Ramanujan, Asymptotic formulae in combinatory analysis, Proc. London Math. Soc M.D. Hirschhorn, J.A. Sellers, Arithmetic relations for overpartitions, J. Combin. Math. Combin. Comput M.D. Hirschhorn, J.A. Sellers, Arithmetic properties of overpartitions into odd parts, Ann. Comb L.K. Hua, On the number of partitions into unequal parts, Trans. Amer. Math. Soc S. Isei, A partition functin with some congruence condition, Amer. J. Math S. Isei, On some partition functions, J. Math. Soc. Japan S. Isei, Partitions in certain arithmetic progressions, Amer. J. Math J. Lehner, A partition function connected with the modulus five, Due Math. J J. Livingood, A partition function with prime modulus p > 3, Amer. J. Math J. Lovejoy, Gordon s theorem for overpartitions, J. Combin. Theory Ser. A J. Lovejoy, Overpartitions and real quadratic fields, J. Number Theory J. Lovejoy, Overpartition theorems of the Rogers-Ramanujan type, J. London Math. Soc J. Lovejoy, A theorem on seven-colored overpartitions and its applications, Int. J. Number Theory J. Lovejoy, Ran and conjugation for the Frobenius representation of an overpartition, Ann. Comb J. Lovejoy, Partitions and overpartitions with attached parts, Arch. Math. Basel K. Mahlburg, The overpartition function modulo small powers of 2, Discrete Math , no. 3, I. Niven, On a certain partition function, Amer. J. Math H. Rademacher, On the partition function pn, Proc. London Math. Soc H. Rademacher, On the epanson of the partition function in a series, Ann. Math H. Rademacher, Lectures on Analytic Number Theory, Tata Institute, Bombay, H. Rademacher, Topics in Analytic Number Theory, Die Grundelhren der mathematischen Wissenschaften, Bd. 69, Springer-Verlag, N. Robbins, Some properties of overpartitions. JP J. Algebra Number Theory Appl Ø. Rødseth, J. A. Sellers, On m-ary overpartitions, Ann. Comb L.J. Slater, Further identities of the Rogers-Ramanujan type, Proc. London Math. Soc A.V. Sills, Identities of the Rogers-Ramanujan-Slater type, Internat. J. Number Theory A.V. Sills, A Rademacher type formula for overpartitions, preprint, A.V. Sills, Towards an automation of the circle method, preprint, M. Tamba and C. Xie, Level three standard modules for A 2 2 and combinatorial identities, J. Pure Appl. Algebra , no., M. Petovše, H.S. Wilf and D. Zeilberger, A B, A.K. Peters, Wellesley, MA, V.V. Subramanyasastri, Partitions with congruence conditions, J. Indian Math. Soc C. Syes writer, producer, director, The Man Who Loved Numbers, NOVA documentary, PBS, WGBH Boston original airdate: March 22, H.S. Wilf and D. Zeilberger, Rational functions certify combinatorial identities, J. Amer. Math. Soc H.S. Wilf and D. Zeilberger, An algorithmic proof theory for hypergeometric ordinary and q multisum/integral identities, Invent. Math D. Zeilberger, A fast algorithm for proving termination hypergeometric identities, Discrete Math D. Zeilberger, The method of creative telescoping, J. Symbolic Comput