NEW IDENTITIES INVOLVING SUMS OF THE TAILS RELATED TO REAL QUADRATIC FIELDS KATHRIN BRINGMANN AND BEN KANE
|
|
- Alvin Pitts
- 5 years ago
- Views:
Transcription
1 NEW IDENTITIES INVOLVING SUMS OF THE TAILS RELATED TO REAL QUADRATIC FIELDS KATHRIN BRINGMANN AND BEN KANE To George Andrews, who has been a great inspiration, on the occasion of his 70th birthday Abstract. In previous work, the authors discovered new examples of q-hypergeometric series related to the arithmetic of Q and Q 3. Building on this work, we construct in this paper sum of the tails identities for which some which some of these functions occur as error terms. As an application, we obtain formulas for the generating function of a certain zeta functions for real quadratic fields at negative integers.. Introduction Andrews, Dyson and Hickerson [] investigated the function σq := + n q nn+ = + q q + q q q n which first appeard in Ramanujan s lost notebook [7]. We denote as usual a n := a; q n := n m=0 aqm. By showing that the Fourier coefficients of σq are connected to the arithmetic of Q 6, the authors of [] were able to prove that σ is lacunary, i.e. its coefficents are almost always zero, and yet attains every integer infinitely many times. Since Andrews, Dyson and Hickerson s investigation, the function σq has shown up in a variety of settings and is related to the theory of automorphic forms in many interesting ways, as described in [4]. Here we consider one such type of result involving sum of tails identities. Such identities were first studied by Zagier [8] in his work on Vassiliev invariants, who showed that q q n = q D q + E q, where D q := + n dnq n, E q := n nq n 4. n Here dn denotes the number of divisors of n and n is the Kronecker symbol. In [3], Andrews, Jimenez-Urroz, and Ono obtained a number of related identites, which have the form. F q F n q = F qdq + Eq, where F is a modular infinite product, F n F, and D is a divisor function. Since the coefficients of E grow much slower than those of F D, the function E may be considered as an 000 Mathematics Subject Classification. P8, E6, 05A7. Key words and phrases. q-hypergeometric series, sum of tails, modular form, real quadratic fields. The first author was partially supported by NSF grant DMS
2 KATHRIN BRINGMANN AND BEN KANE error series. Of particular interest are special cases of. in which the error series is σq, for example the following formula which can be found in Ramanujan s lost notebook. q q n = q D q + σq. In [4], the authors discovered 8 additional examples of q-hypergeometric series related to the arithmetic of Q and Q 3 including the functions fq := q n q n+ q n, hq := q n+ q n+ q n+. Using the theory of Bailey pairs, it was shown in [4] that qfq = 4.3 q N a, N a.4 hq = N a q N a, where K := Q, O K is the ring of integers of K, the sum runs over all ideals of O K, and N denotes the norm of an ideal. As in the case of σ, the identities in.3 and.4 imply arithmetic information about the coefficients of f and h including lacunary behavior. In this note we consider sums of tails identities resembling. which involve the functions f and h as error series. To state our results, for i {0, } we define θ i q := n Z q n+i q A q B q A B, we let its finite com- and, denoting as usual the q-binomial coefficients by [ ] A B := panion be given by [ n + θ i,n q := q n n j ] q j+i. We note that lim n θ i,n q = θ i q and that the theta series θ i q are modular forms which can be expressed as an infinite product, namely they can be written θ 0 q and θ q = q q6 ;q 6 q 8 ;q 8. Moreover, we define D q := d o nq n, n= where d 0 n counts the number of odd divisors of n. = q ;q 5 q q 4 ;q 4 Theorem.. We have the sum of tails identities θ 0 q θ 0,n q θ 0 q.5 q q; q q n q; q = n+ q q; q D q hq, θ q θ,n q θ q.6 q q; q q n q; q = n+ q q; q D q + q fq.
3 NEW IDENTITIES INVOLVING SUMS OF THE TAILS RELATED TO REAL QUADRATIC FIELDS 3 As an application, we next consider formulas for the generating function of the following zeta functions for real quadratic fields at negative integers, paralleling the results obtained in [8] and [3]. For this define the usual Hecke L-function ζ K s for K and a twisted zeta-function ζ K s by ζ K s := 4 N a s, ζk s := N a s. N a Theorem. will give us the following formulas from which ζ K s and ζ K s may be calculated at negative integers. Theorem.. As a power series in t, we have the equations.7.8 θ 0,n e t e t ; e t n e t ; e t = n ζ K n tn n+ n!, n θ,n e t e t ; e t n e t ; e t = n ζk n tn n+ n!. n We note that similarly we could also consider the limiting behavior at other odds roots of unity. This paper is organized as follows. In Section, we first introduce an auxilliary function with an extra parameter, which is in special cases related to f and h, and thus also to the arithmetic of K. This auxiliary function also has some nice combinatorial meaning. We then show that this function is the error series in a sum of tails identity. In Section 3, we use the results from Section to prove Theorem.. We conclude section 3 with another sums of tails result involving h, but where hq is not the error series. Finally, we will establish the formulas for ζ K s and ζ K s at negative integers in Section 4. Acknowledgements The authors thank George Andrews and Jeremy Lovejoy for fruitful conversations. Moreover, they thank Ken Ono for helpful comments on an earlier version of the paper.. An Auxilliary Function We will first investigate the properties of an auxilliary function which we will then bootstrap to obtain the main results for f and h. To this end, we define g z q := q n q n+ z n q n. We will see in equation.6 that z qg z q is the generating function for a natural combinatorial object. For this recall that an overpartition of the integer n is a partition, where the last occurence of each part may be overlined. Denote by P the set of overpartitions into odd parts where the largest part must be overlined. If we denote the largest part by LΛ and the number of parts by MΛ, then the coefficient of z m q n in z qg z q is the number of overpartitions Λ P of n with MΛ = m, weighted by 4 LΛ. We will particularly be interested in the two specializations z = ±. For the specialization z =, the n-th Fourier coefficient of qg + q equals #{Λ P : LΛ mod 4} #{Λ P : LΛ 3 mod 4},
4 4 KATHRIN BRINGMANN AND BEN KANE while the n-th coefficient of qg q equals #{Λ P : LΛ MΛ + mod 4} #{Λ P : LΛ MΛ + mod 4}. However, we will see that the coefficients of qg ± q will always be equal up to absolute value, due to the following relations with fq and hq: g +q + g q = fq, g +q g q = q hq. Taking q q and adding the two above equations relates the coefficients of g q to the arithmetic of O K by. qg + q = qfq + hq = [ ] 4 + N a i N a q N a, N a which follows from equations.3 and.4. Here we have used the fact that there is a unique ideal of O K of norm, so hq is precisely the generating function for the elements of norm n, weighted by i n. Subtracting the two equations yields. qg q = qfq hq = [ ] 4 + N a + i N a q N a. N a Remark. Since 4 n = 0 and + n+ = 0, it follows that the absolute value of the n-th Fourier coefficient of both qg + q and qg q equals the number of ideals of O K of norm n. Hence, although the coefficients of these series are not multiplicative, they are multiplicative up to ±. We next show a sum of the tail-identity involving the functions g z. For this, we let Theorem.. We have zq.3 zq n zq zq n+ zq n D z q := z q n+. n z q n+ zq n This immediately implies the following corollary. Corollary.. The following equations hold q q n.4 q q n+ q q n.5 q q n+ = zq zq D z q + D q z q gz q. = q q D q q g q, = q q D q + q g q. Proof. From Theorem of [3] with a = zq and t = zq, one may easily deduce that zq zq n = zq D n zq zq n zq z q + D q + + z q n n q n. n z q 3 n
5 NEW IDENTITIES INVOLVING SUMS OF THE TAILS RELATED TO REAL QUADRATIC FIELDS 5 Dividing both sides by zq = zq z q and shifting the sum of tails gives zq zq n = zq D zq zq n+ zq z q + D q z q z q n n q n. z q n+ Using Heine s second transformation cf. [, p.,.5] with a = b = q, c = q, and z = z q gives that.6 g z q = z q n z q n+ n q n. This gives the claim. 3. Proof of the Main Result In this section, we will use Theorem. to obtain the desired sum of tails results for fq and hq. Since the proof of both identities are quite similar, we only prove the first here. We first define and its finite companion F q := q + q q q n F n q := q n + q. q n+ q n+ Specializing equation.3 once with z = and once z = and then summing gives F q F n q = F qd q hq. Thus it remains to compute F q and F n q. Using the Jacobi triple product identity cf. [5, p. 7] one obtains F q q = q ; q 4 q ; q q ; q + q ; q 5 q 4 ; q 4 q = q ; q 4 q ; q n Z n q n + q n. n Z Thus To compute F n q we first rewrite F n q = q; q n+ F q = q; q q θ 0 q. q n qn+ + q + q n+ n
6 6 KATHRIN BRINGMANN AND BEN KANE and then use McMahon s finite version of the Jacobi triple product identity [6, vol., Section 33] with x = q for the first summand and x = q for the second summand. This gives F n q = [ ] q; q q n+ n [ ] n+ n + j j q j + + q n+ n q j n + j = [ ] n [ ] q; q q n+ n + j j + q n+ n q n + j + j +j+ = = q; q n+ q; q n+ q n q j q n+j+ + q n+j+ q n j q n+j+ q n j [ ] n + q n j j = q; q θ 0,n τ. n+ q n There is additionally another sum of tails identity related to g z q. In this case g z q does not play the role of the error series but rather as part of the divisor function. Theorem 3.. We have the sum of tails identity q q n q q n q n+ = q q D q hq. Proof. From Theorem of [3] with a = q, b = z q, and c = z q 3, one obtains q q n = q q n q q n z q n+ q q n q n q n q n z n q n n n = q D q z q gz q. q The result then follows after summing the terms with z = and z = and dividing by. In addition to these sum of tails identities, the auxilliary function g z q also allows us to deduce new combinatorial interpretations for fq and hq in terms of overpartitions. Using Heine s first transformation cf. [, p. 0,.] with a = b = q, c = q, and z = z q gives g z q = q q n q q n z q n+ qn. After pulling the infinite product inside the sum, this immediately gives fq = q n+ n q n+ q n+ q n, hq = q n+ n q n+ q n+ q n+. To describe the new combinatorial interpretation for the coefficients of f and h, let n = sλ denote the smallest part of the overpartition Λ. Then fq is the generating function for overpartitions where sλ is overlined and occurs exactly once and sλ+ cannot be overlined, weighted by MΛ.
7 NEW IDENTITIES INVOLVING SUMS OF THE TAILS RELATED TO REAL QUADRATIC FIELDS 7 Similarly, writing n + = n + n +, we see that hq is the generating function for overpartitions for which sλ is overlined and occurs precisely once, a non-overlined part of size sλ + must occur, and sλ + cannot be overlined, again weighted by MΛ. 4. Values of zeta functions at negative integers We will use Theorem. to establish Theorem. here. Proof of Theorem.. This proof will follow analogously to that of Zagier [8, Theorem 3] so we will give a brief argument for.7 and leave the details of.8 to the reader. We first note that θ 0 q q q; q = q q ; q 3 q 4 ; q 4 and hence vanishes to infinite order as q. We next replace q by e t and define an and bn by the asymptotic expansions as t 0 given by θ 0,n e t e t ; e t n e t ; e t = ant n, n+ h e t := bnt n. e tn a By Theorem. we have an = bn for every n 0. We then integrate to obtain h e t t s dt = e tn a t s dt = Γs N a s = Γsζ K s. 0 0 The residue at s = n is hence given by n n! ζ K n. However, for any N fixed we have h e t t s dt = bnt n + Ot N t s dt = bn s + n + Hs, n<n 0 n<n where Hs is holomorphic for Res > N, so that for n < N the residue at s = n is bn. References [] G. Andrews, q-series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra, CBMS Regional Conference Series in Mathematics, no. 66, Amer. Math. Soc., Providence, RI, pp. 30, 986. [] G. Andrews, F. Dyson, and D. Hickerson, Partitions and indefinite quadratic forms, Invent. Math , pages [3] G. Andrews, J. Jimenez-Urroz, K. Ono, q-series identities and values of certain L-functions, Duke Math. J , pages [4] K. Bringmann, B. Kane, Multiplicative q-hypergeometric series arising from real quadratic fields, preprint, arxiv: [5] K. Ono, Web of modularity: Arithmetic of the coefficients of modular forms and q-series, CBMS Regional Conference Series in Mathematics, no. 0, Amer. Math. Soc., Providence, RI, 003. [6] P. A. MacMahon, Combinatory Analysis vol. I, II. Cambridge Univ. Press, 95. Reprinted by Chelsea, New York, 960. [7] S. Ramanujan, The lost notebook and other unpublished papers, Narosa, New Delhi, 988. [8] D. Zagier, Vassiliev invariants and a strange identity related to the Dedekind eta-function, Topology 40 00, pages
8 8 KATHRIN BRINGMANN AND BEN KANE Mathematical Institute, University of Cologne, Weyertal 86-90, 5093 Cologne, Germany address: Math Department, Radboud University, Postbus 900, 6500 GL, Nijmegen, Netherlands address:
q-series IDENTITIES AND VALUES OF CERTAIN L-FUNCTIONS Appearing in the Duke Mathematical Journal.
q-series IDENTITIES AND VALUES OF CERTAIN L-FUNCTIONS George E. Andrews, Jorge Jiménez-Urroz and Ken Ono Appearing in the Duke Mathematical Journal.. Introduction and Statement of Results. As usual, define
More informationSELF-CONJUGATE VECTOR PARTITIONS AND THE PARITY OF THE SPT-FUNCTION
SELF-CONJUGATE VECTOR PARTITIONS AND THE PARITY OF THE SPT-FUNCTION GEORGE E ANDREWS FRANK G GARVAN AND JIE LIANG Abstract Let sptn denote the total number of appearances of the smallest parts in all the
More informationALTERNATING STRANGE FUNCTIONS
ALTERNATING STRANGE FUNCTIONS ROBERT SCHNEIDER Abstract. In this note we consider infinite series similar to the strange function F (q) of Kontsevich studied by Zagier Bryson-Ono-Pitman-Rhoades Bringmann-Folsom-
More informationALTERNATING STRANGE FUNCTIONS
ALTERNATING STRANGE FUNCTIONS ROBERT SCHNEIDER Abstract In this note we consider infinite series similar to the strange function F (q) of Kontsevich studied by Zagier, Bryson-Ono-Pitman-Rhoades, Bringmann-Folsom-
More informationIDENTITIES FOR OVERPARTITIONS WITH EVEN SMALLEST PARTS
IDENTITIES FOR OVERPARTITIONS WITH EVEN SMALLEST PARTS MIN-JOO JANG AND JEREMY LOVEJOY Abstract. We prove several combinatorial identities involving overpartitions whose smallest parts are even. These
More informationMODULAR FORMS ARISING FROM Q(n) AND DYSON S RANK
MODULAR FORMS ARISING FROM Q(n) AND DYSON S RANK MARIA MONKS AND KEN ONO Abstract Let R(w; q) be Dyson s generating function for partition ranks For roots of unity ζ it is known that R(ζ; q) and R(ζ; /q)
More informationTHE IMPORTANCE OF BEING STRANGE
THE IMPORTANCE OF BEING STRANGE ROBERT SCHNEIDER Abstract. We discuss infinite series similar to the strange function F (q) of Kontsevich which has been studied by Zagier Bryson Ono Pitman Rhoades and
More informationA PARTITION IDENTITY AND THE UNIVERSAL MOCK THETA FUNCTION g 2
A PARTITION IDENTITY AND THE UNIVERSAL MOCK THETA FUNCTION g KATHRIN BRINGMANN, JEREMY LOVEJOY, AND KARL MAHLBURG Abstract. We prove analytic and combinatorial identities reminiscent of Schur s classical
More informationMOCK MODULAR FORMS AS p-adic MODULAR FORMS
MOCK MODULAR FORMS AS p-adic MODULAR FORMS KATHRIN BRINGMANN, PAVEL GUERZHOY, AND BEN KANE Abstract. In this paper, we consider the question of correcting mock modular forms in order to obtain p-adic modular
More informationDEDEKIND S ETA-FUNCTION AND ITS TRUNCATED PRODUCTS. George E. Andrews and Ken Ono. February 17, Introduction and Statement of Results
DEDEKIND S ETA-FUNCTION AND ITS TRUNCATED PRODUCTS George E. Andrews and Ken Ono February 7, 2000.. Introduction and Statement of Results Dedekind s eta function ηz, defined by the infinite product ηz
More informationTHE BAILEY TRANSFORM AND FALSE THETA FUNCTIONS
THE BAILEY TRANSFORM AND FALSE THETA FUNCTIONS GEORGE E ANDREWS 1 AND S OLE WARNAAR 2 Abstract An empirical exploration of five of Ramanujan s intriguing false theta function identities leads to unexpected
More information#A22 INTEGERS 17 (2017) NEW CONGRUENCES FOR `-REGULAR OVERPARTITIONS
#A22 INTEGERS 7 (207) NEW CONGRUENCES FOR `-REGULAR OVERPARTITIONS Shane Chern Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania shanechern@psu.edu Received: 0/6/6,
More informationTHE FIRST POSITIVE RANK AND CRANK MOMENTS FOR OVERPARTITIONS
THE FIRST POSITIVE RANK AND CRANK MOMENTS FOR OVERPARTITIONS GEORGE ANDREWS, SONG HENG CHAN, BYUNGCHAN KIM, AND ROBERT OSBURN Abstract. In 2003, Atkin Garvan initiated the study of rank crank moments for
More informationA q-series IDENTITY AND THE ARITHMETIC OF HURWITZ ZETA FUNCTIONS
A -SERIES IDENTITY AND THE ARITHMETIC OF HURWITZ ZETA FUNCTIONS GWYNNETH H COOGAN AND KEN ONO Introduction and Statement of Results In a recent paper [?], D Zagier used a -series identity to prove that
More informationarxiv: v2 [math.co] 9 May 2017
Partition-theoretic formulas for arithmetic densities Ken Ono, Robert Schneider, and Ian Wagner arxiv:1704.06636v2 [math.co] 9 May 2017 In celebration of Krishnaswami Alladi s 60th birthday. Abstract If
More information4-Shadows in q-series and the Kimberling Index
4-Shadows in q-series and the Kimberling Index By George E. Andrews May 5, 206 Abstract An elementary method in q-series, the method of 4-shadows, is introduced and applied to several poblems in q-series
More informationGUO-NIU HAN AND KEN ONO
HOOK LENGTHS AND 3-CORES GUO-NIU HAN AND KEN ONO Abstract. Recently, the first author generalized a formula of Nekrasov and Okounkov which gives a combinatorial formula, in terms of hook lengths of partitions,
More informationMOCK THETA FUNCTIONS AND THETA FUNCTIONS. Bhaskar Srivastava
NEW ZEALAND JOURNAL OF MATHEMATICS Volume 36 (2007), 287 294 MOCK THETA FUNCTIONS AND THETA FUNCTIONS Bhaskar Srivastava (Received August 2004). Introduction In his last letter to Hardy, Ramanujan gave
More informationOVERPARTITION M 2-RANK DIFFERENCES, CLASS NUMBER RELATIONS, AND VECTOR-VALUED MOCK EISENSTEIN SERIES
OVERPARTITION M -RANK DIFFERENCES, CLASS NUMBER RELATIONS, AND VECTOR-VALUED MOCK EISENSTEIN SERIES BRANDON WILLIAMS Abstract. We prove that the generating function of overpartition M-rank differences
More informationPARTITION-THEORETIC FORMULAS FOR ARITHMETIC DENSITIES
PARTITION-THEORETIC FORMULAS FOR ARITHMETIC DENSITIES KEN ONO, ROBERT SCHNEIDER, AND IAN WAGNER In celebration of Krishnaswami Alladi s 60th birthday Abstract If gcd(r, t) = 1, then a theorem of Alladi
More informationarxiv: v1 [math.co] 25 Nov 2018
The Unimodality of the Crank on Overpartitions Wenston J.T. Zang and Helen W.J. Zhang 2 arxiv:8.003v [math.co] 25 Nov 208 Institute of Advanced Study of Mathematics Harbin Institute of Technology, Heilongjiang
More informationARITHMETIC OF THE 13-REGULAR PARTITION FUNCTION MODULO 3
ARITHMETIC OF THE 13-REGULAR PARTITION FUNCTION MODULO 3 JOHN J WEBB Abstract. Let b 13 n) denote the number of 13-regular partitions of n. We study in this paper the behavior of b 13 n) modulo 3 where
More informationOVERPARTITIONS AND GENERATING FUNCTIONS FOR GENERALIZED FROBENIUS PARTITIONS
OVERPARTITIONS AND GENERATING FUNCTIONS FOR GENERALIZED FROBENIUS PARTITIONS SYLVIE CORTEEL JEREMY LOVEJOY AND AE JA YEE Abstract. Generalized Frobenius partitions or F -partitions have recently played
More informationarxiv: v2 [math.nt] 9 Apr 2015
CONGRUENCES FOR PARTITION PAIRS WITH CONDITIONS arxiv:408506v2 mathnt 9 Apr 205 CHRIS JENNINGS-SHAFFER Abstract We prove congruences for the number of partition pairs π,π 2 such that π is nonempty, sπ
More informationMock and quantum modular forms
Mock and quantum modular forms Amanda Folsom (Amherst College) 1 Ramanujan s mock theta functions 2 Ramanujan s mock theta functions 1887-1920 3 Ramanujan s mock theta functions 1887-1920 4 History S.
More informationCongruences for Fishburn numbers modulo prime powers
Congruences for Fishburn numbers modulo prime powers Armin Straub Department of Mathematics University of Illinois at Urbana-Champaign July 16, 2014 Abstract The Fishburn numbers ξ(n) are defined by the
More informationDIVISIBILITY PROPERTIES OF THE 5-REGULAR AND 13-REGULAR PARTITION FUNCTIONS
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (008), #A60 DIVISIBILITY PROPERTIES OF THE 5-REGULAR AND 13-REGULAR PARTITION FUNCTIONS Neil Calkin Department of Mathematical Sciences, Clemson
More informationArithmetic properties of overcubic partition pairs
Arithmetic properties of overcubic partition pairs Bernard L.S. Lin School of Sciences Jimei University Xiamen 3101, P.R. China linlsjmu@13.com Submitted: May 5, 014; Accepted: Aug 7, 014; Published: Sep
More informationASYMPTOTICS FOR RANK AND CRANK MOMENTS
ASYMPTOTICS FOR RANK AND CRANK MOMENTS KATHRIN BRINGMANN, KARL MAHLBURG, AND ROBERT C. RHOADES Abstract. Moments of the partition rank and crank statistics have been studied for their connections to combinatorial
More informationSOME CONGRUENCES FOR PARTITION FUNCTIONS RELATED TO MOCK THETA FUNCTIONS ω(q) AND ν(q) S.N. Fathima and Utpal Pore (Received October 13, 2017)
NEW ZEALAND JOURNAL OF MATHEMATICS Volume 47 2017), 161-168 SOME CONGRUENCES FOR PARTITION FUNCTIONS RELATED TO MOCK THETA FUNCTIONS ωq) AND νq) S.N. Fathima and Utpal Pore Received October 1, 2017) Abstract.
More informationTHE NUMBER OF PARTITIONS INTO DISTINCT PARTS MODULO POWERS OF 5
THE NUMBER OF PARTITIONS INTO DISTINCT PARTS MODULO POWERS OF 5 JEREMY LOVEJOY Abstract. We establish a relationship between the factorization of n+1 and the 5-divisibility of Q(n, where Q(n is the number
More informationDIVISIBILITY AND DISTRIBUTION OF PARTITIONS INTO DISTINCT PARTS
DIVISIBILITY AND DISTRIBUTION OF PARTITIONS INTO DISTINCT PARTS JEREMY LOVEJOY Abstract. We study the generating function for (n), the number of partitions of a natural number n into distinct parts. Using
More informationOn an identity of Gessel and Stanton and the new little Göllnitz identities
On an identity of Gessel and Stanton and the new little Göllnitz identities Carla D. Savage Dept. of Computer Science N. C. State University, Box 8206 Raleigh, NC 27695, USA savage@csc.ncsu.edu Andrew
More informationCONGRUENCES FOR BROKEN k-diamond PARTITIONS
CONGRUENCES FOR BROKEN k-diamond PARTITIONS MARIE JAMESON Abstract. We prove two conjectures of Paule and Radu from their recent paper on broken k-diamond partitions. 1. Introduction and Statement of Results
More informationPARITY OF THE COEFFICIENTS OF KLEIN S j-function
PARITY OF THE COEFFICIENTS OF KLEIN S j-function CLAUDIA ALFES Abstract. Klein s j-function is one of the most fundamental modular functions in number theory. However, not much is known about the parity
More informationCONGRUENCES MODULO SQUARES OF PRIMES FOR FU S k DOTS BRACELET PARTITIONS
CONGRUENCES MODULO SQUARES OF PRIMES FOR FU S k DOTS BRACELET PARTITIONS CRISTIAN-SILVIU RADU AND JAMES A SELLERS Dedicated to George Andrews on the occasion of his 75th birthday Abstract In 2007, Andrews
More information= (q) M+N (q) M (q) N
A OVERPARTITIO AALOGUE OF THE -BIOMIAL COEFFICIETS JEHAE DOUSSE AD BYUGCHA KIM Abstract We define an overpartition analogue of Gaussian polynomials (also known as -binomial coefficients) as a generating
More informationCOMBINATORICS OF RAMANUJAN-SLATER TYPE IDENTITIES
COMBINATORICS OF RAMANUJAN-SLATER TYPE IDENTITIES James McLaughlin Department of Mathematics, West Chester University, West Chester, PA 9383, USA jmclaughl@wcupa.edu Andrew V. Sills Department of Mathematical
More informationRamanujan-Slater Type Identities Related to the Moduli 18 and 24
Ramanujan-Slater Type Identities Related to the Moduli 18 and 24 James McLaughlin Department of Mathematics, West Chester University, West Chester, PA; telephone 610-738-0585; fax 610-738-0578 Andrew V.
More informationInteger Partitions With Even Parts Below Odd Parts and the Mock Theta Functions
Integer Partitions With Even Parts Below Odd Parts and the Mock Theta Functions by George E. Andrews Key Words: Partitions, mock theta functions, crank AMS Classification Numbers: P84, P83, P8, 33D5 Abstract
More informationRAMANUJAN S LOST NOTEBOOK: COMBINATORIAL PROOFS OF IDENTITIES ASSOCIATED WITH HEINE S TRANSFORMATION OR PARTIAL THETA FUNCTIONS
RAMANUJAN S LOST NOTEBOOK: COMBINATORIAL PROOFS OF IDENTITIES ASSOCIATED WITH HEINE S TRANSFORMATION OR PARTIAL THETA FUNCTIONS BRUCE C. BERNDT, BYUNGCHAN KIM, AND AE JA YEE Abstract. Combinatorial proofs
More informationCOMBINATORICS OF RAMANUJAN-SLATER TYPE IDENTITIES. James McLaughlin Department of Mathematics, West Chester University, West Chester, PA 19383, USA
COMBINATORICS OF RAMANUJAN-SLATER TYPE IDENTITIES James McLaughlin Department of Mathematics, West Chester University, West Chester, PA 9383, USA jmclaughl@wcupa.edu Andrew V. Sills Department of Mathematical
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics ELLIPTIC FUNCTIONS TO THE QUINTIC BASE HENG HUAT CHAN AND ZHI-GUO LIU Volume 226 No. July 2006 PACIFIC JOURNAL OF MATHEMATICS Vol. 226, No., 2006 ELLIPTIC FUNCTIONS TO THE
More informationCONGRUENCES FOR POWERS OF THE PARTITION FUNCTION
CONGRUENCES FOR POWERS OF THE PARTITION FUNCTION MADELINE LOCUS AND IAN WAGNER Abstract. Let p tn denote the number of partitions of n into t colors. In analogy with Ramanujan s work on the partition function,
More informationNew congruences for overcubic partition pairs
New congruences for overcubic partition pairs M. S. Mahadeva Naika C. Shivashankar Department of Mathematics, Bangalore University, Central College Campus, Bangalore-560 00, Karnataka, India Department
More informationTATE-SHAFAREVICH GROUPS OF THE CONGRUENT NUMBER ELLIPTIC CURVES. Ken Ono. X(E N ) is a simple
TATE-SHAFAREVICH GROUPS OF THE CONGRUENT NUMBER ELLIPTIC CURVES Ken Ono Abstract. Using elliptic modular functions, Kronecker proved a number of recurrence relations for suitable class numbers of positive
More informationPARITY OF THE PARTITION FUNCTION. (Communicated by Don Zagier)
ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 1, Issue 1, 1995 PARITY OF THE PARTITION FUNCTION KEN ONO (Communicated by Don Zagier) Abstract. Let p(n) denote the number
More informationNew infinite families of congruences modulo 8 for partitions with even parts distinct
New infinite families of congruences modulo for partitions with even parts distinct Ernest X.W. Xia Department of Mathematics Jiangsu University Zhenjiang, Jiangsu 212013, P. R. China ernestxwxia@13.com
More informationSingular Overpartitions
Singular Overpartitions George E. Andrews Dedicated to the memory of Paul Bateman and Heini Halberstam. Abstract The object in this paper is to present a general theorem for overpartitions analogous to
More informationq GAUSS SUMMATION VIA RAMANUJAN AND COMBINATORICS
q GAUSS SUMMATION VIA RAMANUJAN AND COMBINATORICS BRUCE C. BERNDT 1 and AE JA YEE 1. Introduction Recall that the q-gauss summation theorem is given by (a; q) n (b; q) ( n c ) n (c/a; q) (c/b; q) =, (1.1)
More informationETA-QUOTIENTS AND ELLIPTIC CURVES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 11, November 1997, Pages 3169 3176 S 0002-9939(97)03928-2 ETA-QUOTIENTS AND ELLIPTIC CURVES YVES MARTIN AND KEN ONO (Communicated by
More informationSOME IDENTITIES RELATING MOCK THETA FUNCTIONS WHICH ARE DERIVED FROM DENOMINATOR IDENTITY
Math J Okayama Univ 51 (2009, 121 131 SOME IDENTITIES RELATING MOCK THETA FUNCTIONS WHICH ARE DERIVED FROM DENOMINATOR IDENTITY Yukari SANADA Abstract We show that there exists a new connection between
More informationCONGRUENCES RELATED TO THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ω(q) AND ν(q)
CONGRUENCES RELATED TO THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ωq) AND νq) GEORGE E. ANDREWS, DONNY PASSARY, JAMES A. SELLERS, AND AE JA YEE Abstract. Recently, Andrews, Dixit and Yee introduced partition
More informationRAMANUJAN S LOST NOTEBOOK: COMBINATORIAL PROOFS OF IDENTITIES ASSOCIATED WITH HEINE S TRANSFORMATION OR PARTIAL THETA FUNCTIONS
RAMANUJAN S LOST NOTEBOOK: COMBINATORIAL PROOFS OF IDENTITIES ASSOCIATED WITH HEINE S TRANSFORMATION OR PARTIAL THETA FUNCTIONS BRUCE C. BERNDT, BYUNGCHAN KIM, AND AE JA YEE 2 Abstract. Combinatorial proofs
More informationCHIRANJIT RAY AND RUPAM BARMAN
ON ANDREWS INTEGER PARTITIONS WITH EVEN PARTS BELOW ODD PARTS arxiv:1812.08702v1 [math.nt] 20 Dec 2018 CHIRANJIT RAY AND RUPAM BARMAN Abstract. Recently, Andrews defined a partition function EOn) which
More informationarxiv: v2 [math.co] 3 May 2016
VARIATION ON A THEME OF NATHAN FINE NEW WEIGHTED PARTITION IDENTITIES arxiv:16050091v [mathco] 3 May 016 ALEXANDER BERKOVICH AND ALI KEMAL UNCU Dedicated to our friend Krishna Alladi on his 60th birthday
More informationPolynomial analogues of Ramanujan congruences for Han s hooklength formula
Polynomial analogues of Ramanujan congruences for Han s hooklength formula William J. Keith CELC, University of Lisbon Email: william.keith@gmail.com Detailed arxiv preprint: 1109.1236 Context Partition
More informationOn the Ordinary and Signed Göllnitz-Gordon Partitions
On the Ordinary and Signed Göllnitz-Gordon Partitions Andrew V. Sills Department of Mathematical Sciences Georgia Southern University Statesboro, Georgia, USA asills@georgiasouthern.edu Version of October
More informationRANK DIFFERENCES FOR OVERPARTITIONS
RANK DIFFERENCES FOR OVERPARTITIONS JEREMY LOVEJOY AND ROBERT OSBURN Abstract In 1954 Atkin Swinnerton-Dyer proved Dyson s conjectures on the rank of a partition by establishing formulas for the generating
More informationRANK AND CONGRUENCES FOR OVERPARTITION PAIRS
RANK AND CONGRUENCES FOR OVERPARTITION PAIRS KATHRIN BRINGMANN AND JEREMY LOVEJOY Abstract. The rank of an overpartition pair is a generalization of Dyson s rank of a partition. The purpose of this paper
More informationCongruences for Fishburn numbers modulo prime powers
Congruences for Fishburn numbers modulo prime powers Partitions, q-series, and modular forms AMS Joint Mathematics Meetings, San Antonio January, 205 University of Illinois at Urbana Champaign ξ(3) = 5
More informationarxiv: v1 [math.nt] 7 Oct 2009
Congruences for the Number of Cubic Partitions Derived from Modular Forms arxiv:0910.1263v1 [math.nt] 7 Oct 2009 William Y.C. Chen 1 and Bernard L.S. Lin 2 Center for Combinatorics, LPMC-TJKLC Nankai University,
More informationOn a certain vector crank modulo 7
On a certain vector crank modulo 7 Michael D Hirschhorn School of Mathematics and Statistics University of New South Wales Sydney, NSW, 2052, Australia mhirschhorn@unsweduau Pee Choon Toh Mathematics &
More informationON A MODULARITY CONJECTURE OF ANDREWS, DIXIT, SCHULTZ, AND YEE FOR A VARIATION OF RAMANUJAN S ω(q)
ON A MODULARITY CONJECTURE OF ANDREWS, DIXIT, SCHULTZ, AND YEE FOR A VARIATION OF RAMANUJAN S ωq KATHRIN BRINGMANN, CHRIS JENNINGS-SHAFFER, AND KARL MAHLBURG Abstract We analyze the mock modular behavior
More informationCONGRUENCES MODULO 2 FOR CERTAIN PARTITION FUNCTIONS
Bull. Aust. Math. Soc. 9 2016, 400 409 doi:10.1017/s000497271500167 CONGRUENCES MODULO 2 FOR CERTAIN PARTITION FUNCTIONS M. S. MAHADEVA NAIKA, B. HEMANTHKUMAR H. S. SUMANTH BHARADWAJ Received 9 August
More informationINDEFINITE THETA FUNCTIONS OF TYPE (n, 1) I: DEFINITIONS AND EXAMPLES
INDEFINITE THETA FUNCTIONS OF TYPE (n, ) I: DEFINITIONS AND EXAMPLES LARRY ROLEN. Classical theta functions Theta functions are classical examples of modular forms which play many roles in number theory
More informationCOMBINATORIAL APPLICATIONS OF MÖBIUS INVERSION
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 COMBINATORIAL APPLICATIONS OF MÖBIUS INVERSION MARIE JAMESON AND ROBERT P. SCHNEIDER (Communicated
More informationArithmetic Properties for Ramanujan s φ function
Arithmetic Properties for Ramanujan s φ function Ernest X.W. Xia Jiangsu University ernestxwxia@163.com Nankai University Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function
More informationA P-ADIC SUPERCONGRUENCE CONJECTURE OF VAN HAMME
A P-ADIC SUPERCONGRUENCE CONJECTURE OF VAN HAMME ERIC MORTENSON Abstract. In this paper we prove a p-adic supercongruence conjecture of van Hamme by placing it in the context of the Beuers-lie supercongruences
More informationCONGRUENCES RELATED TO THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ω(q) AND ν(q)
CONGRUENCES RELATED TO THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ωq) AND νq) GEORGE E. ANDREWS, DONNY PASSARY, JAMES A. SELLERS, AND AE JA YEE Abstract. Recently, Andrews, Dixit, and Yee introduced partition
More informationA Motivated Introduction to Modular Forms
May 3, 2006 Outline of talk: I. Motivating questions II. Ramanujan s τ function III. Theta Series IV. Congruent Number Problem V. My Research Old Questions... What can you say about the coefficients of
More informationMOCK THETA FUNCTIONS OF ORDER 2 AND THEIR SHADOW COMPUTATIONS
MOCK THETA FUNCTIONS OF ORDER AND THEIR SHADOW COMPUTATIONS SOON-YI KANG AND HOLLY SWISHER Abstract Zwegers showed that a mock theta function can be completed to form essentially a real analytic modular
More informationarxiv: v1 [math.nt] 10 Jan 2015
DISTINCT PARTS PARTITIONS WITHOUT SEQUENCES arxiv:150102305v1 [mathnt] 10 Jan 2015 KATHRIN BRINGMANN, KARL MAHLBURG, AND KARTHIK NATARAJ Abstract Partitions without sequences of consecutive integers as
More informationRESEARCH STATEMENT ROBERT SCHNEIDER
RESEARCH STATEMENT ROBERT SCHNEIDER My primary research interest is in systematizing number theory at the intersection of its additive and multiplicative branches, and proving applications in partitions,
More informationRamanujan-type congruences for broken 2-diamond partitions modulo 3
Progress of Projects Supported by NSFC. ARTICLES. SCIENCE CHINA Mathematics doi: 10.1007/s11425-014-4846-7 Ramanujan-type congruences for broken 2-diamond partitions modulo 3 CHEN William Y.C. 1, FAN Anna
More informationTHE ARITHMETIC OF BORCHERDS EXPONENTS. Jan H. Bruinier and Ken Ono
THE ARITHMETIC OF BORCHERDS EXPONENTS Jan H. Bruinier and Ken Ono. Introduction and Statement of Results. Recently, Borcherds [B] provided a striking description for the exponents in the naive infinite
More informationRamanujan s last prophecy: quantum modular forms
Ramanujan s last prophecy: quantum modular forms Ken Ono (Emory University) Introduction Death bed letter Dear Hardy, I am extremely sorry for not writing you a single letter up to now. I discovered very
More informationModular forms, combinatorially and otherwise
Modular forms, combinatorially and otherwise p. 1/103 Modular forms, combinatorially and otherwise David Penniston Sums of squares Modular forms, combinatorially and otherwise p. 2/103 Modular forms, combinatorially
More informationThe Bhargava-Adiga Summation and Partitions
The Bhargava-Adiga Summation and Partitions By George E. Andrews September 12, 2016 Abstract The Bhargava-Adiga summation rivals the 1 ψ 1 summation of Ramanujan in elegance. This paper is devoted to two
More informationOn the expansion of Ramanujan's continued fraction of order sixteen
Tamsui Oxford Journal of Information and Mathematical Sciences 31(1) (2017) 81-99 Aletheia University On the expansion of Ramanujan's continued fraction of order sixteen A. Vanitha y Department of Mathematics,
More informationArithmetic Relations for Overpartitions
Arithmetic Relations for Overpartitions Michael D. Hirschhorn School of Mathematics, UNSW, Sydney 2052, Australia m.hirschhorn@unsw.edu.au James A. Sellers Department of Mathematics The Pennsylvania State
More informationASYMPTOTIC FORMULAS FOR STACKS AND UNIMODAL SEQUENCES
ASYMPTOTIC FORMULAS FOR STACKS AND UNIMODAL SEQUENCES KATHRIN BRINGMANN AND KARL MAHLBURG Abstract. We study enumeration functions for unimodal sequences of positive integers, where the size of a sequence
More informationRamanujan-type Congruences for Broken 2-Diamond Partitions Modulo 3
Ramanujan-type Congruences for Broken 2-Diamond Partitions Modulo 3 William Y.C. Chen 1, Anna R.B. Fan 2 and Rebecca T. Yu 3 1,2,3 Center for Combinatorics, LPMC-TJKLC Nankai University, Tianjin 300071,
More informationRamanujan-type congruences for overpartitions modulo 16. Nankai University, Tianjin , P. R. China
Ramanujan-type congruences for overpartitions modulo 16 William Y.C. Chen 1,2, Qing-Hu Hou 2, Lisa H. Sun 1,2 and Li Zhang 1 1 Center for Combinatorics, LPMC-TJKLC Nankai University, Tianjin 300071, P.
More informationPartitions With Parts Separated By Parity
Partitions With Parts Separated By Parity by George E. Andrews Key Words: partitions, parity of parts, Ramanujan AMS Classification Numbers: P84, P83, P8 Abstract There have been a number of papers on
More informationREGULARIZED INNER PRODUCTS AND WEAKLY HOLOMORPHIC HECKE EIGENFORMS
REGULARIZED INNER PRODUCTS AND WEAKLY HOLOMORPHIC HECKE EIGENFORMS KATHRIN BRINGMANN AND BEN KANE 1. Introduction and statement of results For κ Z, denote by M 2κ! the space of weight 2κ weakly holomorphic
More informationOn m-ary Overpartitions
On m-ary Overpartitions Øystein J. Rødseth Department of Mathematics, University of Bergen, Johs. Brunsgt. 1, N 5008 Bergen, Norway E-mail: rodseth@mi.uib.no James A. Sellers Department of Mathematics,
More informationA PERIODIC APPROACH TO PLANE PARTITION CONGRUENCES
A PERIODIC APPROACH TO PLANE PARTITION CONGRUENCES MATTHEW S. MIZUHARA, JAMES A. SELLERS, AND HOLLY SWISHER Abstract. Ramanujan s celebrated congruences of the partition function p(n have inspired a vast
More informationEXACT FORMULAS FOR COEFFICIENTS OF JACOBI FORMS
EXACT FORMULAS FOR COEFFICIENTS OF JACOBI FORMS KATHRIN BRINGMANN AND OLAV K. RICHTER Abstract. In previous work, we introduced harmonic Maass-Jacobi forms. The space of such forms includes the classical
More informationINFINITE FAMILIES OF STRANGE PARTITION CONGRUENCES FOR BROKEN 2-DIAMONDS
December 1, 2009 INFINITE FAMILIES OF STRANGE PARTITION CONGRUENCES FOR BROKEN 2-DIAMONDS PETER PAULE AND SILVIU RADU Dedicated to our friend George E. Andrews on the occasion of his 70th birthday Abstract.
More informationON THE POSITIVITY OF THE NUMBER OF t CORE PARTITIONS. Ken Ono. 1. Introduction
ON THE POSITIVITY OF THE NUMBER OF t CORE PARTITIONS Ken Ono Abstract. A partition of a positive integer n is a nonincreasing sequence of positive integers whose sum is n. A Ferrers graph represents a
More informationFOURIER COEFFICIENTS OF VECTOR-VALUED MODULAR FORMS OF DIMENSION 2
FOURIER COEFFICIENTS OF VECTOR-VALUED MODULAR FORMS OF DIMENSION 2 CAMERON FRANC AND GEOFFREY MASON Abstract. We prove the following Theorem. Suppose that F = (f 1, f 2 ) is a 2-dimensional vector-valued
More informationCOMBINATORIAL PROOFS OF RAMANUJAN S 1 ψ 1 SUMMATION AND THE q-gauss SUMMATION
COMBINATORIAL PROOFS OF RAMANUJAN S 1 ψ 1 SUMMATION AND THE q-gauss SUMMATION AE JA YEE 1 Abstract. Theorems in the theory of partitions are closely related to basic hypergeometric series. Some identities
More informationANALOGUES OF THE TRIPLE PRODUCT IDENTITY, LEBESGUE S IDENTITY AND EULER S PENTAGONAL NUMBER THEOREM
q-hypergeometric PROOFS OF POLYNOMIAL ANALOGUES OF THE TRIPLE PRODUCT IDENTITY, LEBESGUE S IDENTITY AND EULER S PENTAGONAL NUMBER THEOREM S OLE WARNAAR Abstract We present alternative, q-hypergeometric
More informationQuadratic Forms and Congruences for l-regular Partitions Modulo 3, 5 and 7. Tianjin University, Tianjin , P. R. China
Quadratic Forms and Congruences for l-regular Partitions Modulo 3, 5 and 7 Qing-Hu Hou a, Lisa H. Sun b and Li Zhang b a Center for Applied Mathematics Tianjin University, Tianjin 30007, P. R. China b
More informationSOME CONGRUENCES FOR TRACES OF SINGULAR MODULI
SOME CONGRUENCES FOR TRACES OF SINGULAR MODULI P. GUERZHOY Abstract. We address a question posed by Ono [7, Problem 7.30], prove a general result for powers of an arbitrary prime, and provide an explanation
More informationREPRESENTATIONS OF INTEGERS AS SUMS OF SQUARES. Ken Ono. Dedicated to the memory of Robert Rankin.
REPRESENTATIONS OF INTEGERS AS SUMS OF SQUARES Ken Ono Dedicated to the memory of Robert Rankin.. Introduction and Statement of Results. If s is a positive integer, then let rs; n denote the number of
More informationElementary proofs of congruences for the cubic and overcubic partition functions
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 602) 204), Pages 9 97 Elementary proofs of congruences for the cubic and overcubic partition functions James A. Sellers Department of Mathematics Penn State
More informationQUANTUM MODULARITY OF MOCK THETA FUNCTIONS OF ORDER 2. Soon-Yi Kang
Korean J. Math. 25 (2017) No. 1 pp. 87 97 https://doi.org/10.11568/kjm.2017.25.1.87 QUANTUM MODULARITY OF MOCK THETA FUNCTIONS OF ORDER 2 Soon-Yi Kang Abstract. In [9] we computed shadows of the second
More informationGuo, He. November 21, 2015
Math 702 Enumerative Combinatorics Project: Introduction to a combinatorial proof of the Rogers-Ramanujan and Schur identities and an application of Rogers-Ramanujan identity Guo, He November 2, 205 Abstract
More information