COMPUTATIONAL PROOFS OF CONGRUENCES FOR 2-COLORED FROBENIUS PARTITIONS

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1 IJMMS 29: PII. S Hindawi Publishing Corp. COMPUTATIONAL PROOFS OF CONGRUENCES FOR 2-COLORED FROBENIUS PARTITIONS DENNIS EICHHORN and JAMES A. SELLERS Received 13 April 2001 In 1994, the following infinite family of congruences was conjectured for the partition function n which counts the number of 2-colored Frobenius partitions of n: forall n 0andα 1, α,whereλ α is the least positive reciprocal of 12 modulo 5 α. In this paper, the first four cases of this family are proved Mathematics Subject Classification: 05A17, 11P Background and introduction. In his 1984 Memoir of the American Mathematical Society, Andrews [2] introduced two families of partition functions, φ k m and cφ k m, which he called generalized Frobenius partition functions. In this paper, we will focus our attention on one of these functions, namely m, which denotes the number of generalized Frobenius partitions of m with 2 colors. In [2], Andrews gives the generating function for m: mq m = m 0 q 2 ;q 4 q;q 2 4 q4 ;q 4, 1.1 where a;b = 1 a1 ab1 ab 2 1 ab 3. Andrews then proves the following: for all n 0, 5n+3, 1.2 2n+1 mod More recently, Sellers [9] conjectured the following infinite family of congruences satisfied by. Conjecture 1.1. For all n 0 and α 1, α, 1.4 where λ α is the least positive reciprocal of 12 modulo 5 α. The case α = 1is1.2. The reader will note the similarity of this conjecture to the well-known family of congruences for pm, the classical partition function of m: foralln 0, p 5 α n+γ α α, 1.5

2 334 D. EICHHORN AND J. A. SELLERS where γ α is the least positive reciprocal of 24 modulo 5 α. For two different proofs of 1.5, see [1, 6]. Unfortunately, 1.4 has proven to be much more difficult to prove than 1.5. The goal of this paper is to prove the following theorem. Theorem 1.2. For all n 0 and α = 1,2,3,4, where λ α is the least positive reciprocal of 12 modulo 5 α. α, 1.6 In order to prove this theorem, we implement a finitization technique developed recently cf. [3]. In essence, we prove that, for fixed α, if and only if α n 1.7 α n Cα, 1.8 where Cα is an explicit constant dependent on α. We then compute all values of needed to utilize the equivalence above. The development of Cα requires the theory of modular forms as outlined below. 2. Determination of Cα. In this section, we use the theory of modular forms to determine the constant Cα. We do so by constructing a modular form whose Fourier coefficients inherit the congruence properties modulo 5 α of in the desired arithmetic progression. Then, thanks to a theorem of Sturm [10], we will be able to provide explicitly a constant Cα such that if a congruence for the Fourier coefficients of our modular form or equivalently, for holds for all n Cα, the congruence must hold for all n. For a general introduction to the theory of modular forms, see [7]. For an exposition focused on the results we use below, see [3]. We now state Sturm s theorem [10]. Theorem 2.1 Sturm. If fz = anq n and gz = bnq n are holomorphic modular forms of weight k with respect to some congruence subgroup Γ of SL 2 Z with integer coefficients, then fz gzmodl where l is prime if and only if where Ord l Fq := min{n An 0modl}. k [ Ord l fz gz > SL2 Z : Γ ], Sturm s theorem also holds when the prime l is replaced by 5 α, or in fact by any positive integer. Thus, when we let gz = 0, Sturm s theorem allows us to determine when the coefficients an of a holomorphic modular form have the property that an α for all n. We are now ready to state the main result needed to prove Theorem 1.2.

3 COMPUTATIONAL PROOFS OF CONGRUENCES Theorem 2.2. Suppose that α is a positive integer, and let Cα := 6 b 1+4ε 5 α 1 b 5 α 1, where b = bα is the smallest integer greater than 4 5 α 2 with b 5 α mod12, ε = εα = 1 if α is odd, and ε = εα = 2 if α is even. Then if and only if α n 2.3 α n Cα, 2.4 where λ α is the least positive reciprocal of 12 modulo 5 α. Proof. Let fz= η 5 2z η 4 zη 2 4z ηb 2 5 α z η 5 z η5z ε 5 α 1 = anq n, 2.5 where ηz is the Dedekind eta-function, defined by ηz = q 1/24 q;q, q = e 2πiz, b = bα is the smallest integer greater than 4 5 α 2 with b 5 α mod12, ε = εα = 1 if α is odd, and ε = εα = 2ifα is even. Using results from [4, Theorems 3 and 5] on the properties of η-products, we find that fzis a holomorphic modular form of weight b 1/2+2ε 5 α 1 and character χ 0, the trivial character, with respect to Γ α. Notice that ε 5 η 5 α 1 z = 1+5 α hnq n, 2.6 η5z n=1 where the hn are integers, and thus the Fourier coefficients of fz are congruent to the Fourier coefficients of η 5 2z η 4 zη 2 4z ηb 2 5 α z α. 2.7 Next, note that in terms of eta-functions, Thus, if we let nq n = q 1/12 η 5 2z η 4 zη 2 4z. 2.8 q 2b 5α /24 η b 2 5 α z = d 2 5 α n q 2 5αn, 2.9 then a + 2b 5α 2 d 2 5 α m 2 5 α m α m=0

4 336 D. EICHHORN AND J. A. SELLERS Since d0 = 1, this becomes a + 2b 5α 2 5 α n+λ α 24 + d 2 5 α m 2 5 α m α. m= By induction, it is easy to see that α for all n Cα if and only if a +2b 5 α 2/24 α for all n Cα. Hence, we also have that α for all n if and only if a +2b 5 α 2/24 0 α for all n. Now notice that λ α +2b 5 α 2/24 α by hypothesis, so consider f 1 z = fz T 5 α = a 5 α n q n, 2.12 which is also a holomorphic modular form of weight b 1/2+2ε 5 α 1 and character χ 0 with respect to Γ α.see[7, pages ] for a full explanation of the action of the Hecke operators T p. We find by Sturm s theorem that a5 α n α for all n if and only if a 5 α n α b 1/2+2ε 5 α α n 12 p p Therefore, 5 α n + λ α α for all n if and only if the congruence holds for all n Cα. For certain values of α, it is not difficult to make modest improvements to Theorem 1.2. In the case α = 4, this modest improvement will bring Cα more comfortably within the realm of computational feasibility. Theorem 2.3. Let C4 := Then 625n+573 mod625 n 2.15 if and only if 625n+573 mod625 n C Proof. Let fz= η5 2z η η 4 zη 2 4z η44 625zη zη 10 5 z z = anq n, 2.17 η5z where q = e 2πiz. We find that fz is a holomorphic modular form of weight 530 and character χ 0, the trivial character, with respect to Γ

5 COMPUTATIONAL PROOFS OF CONGRUENCES Notice that η 5 z 250 = η5z hnq n, 2.18 where the hn are integers, and thus the Fourier coefficients of fz are congruent to the Fourier coefficients of Recalling that if we let then a n=1 η 5 2z η 4 zη 2 4z η44 625zη zη zmod nq n = q 1/12 η 5 2z η 4 zη 2 4z, 2.20 q 61250/24 η zη zη z = 625n Since d0 = 1, this becomes d625nq 625n, 2.21 d625m 625n mmod625. m= a625n n d625m 625n mmod625. m= By induction, it is easy to see that 625n mod625 for all n C4 if and only if a625n mod625 for all n C4. Hence, we also have that 625n mod625 for all n if and only if a625n mod625 for all n. Now notice that mod625, so consider f 1 z = fz T 625 = a625nq n, 2.24 which is also a holomorphic modular form of weight 530 and character χ 0 with respect to Γ We find by Sturm s theorem that a625n mod625 for all n if and only if a625n mod625 n p p Therefore, 625n+573 mod625 for all n if and only if the congruence holds for all n C4.

6 338 D. EICHHORN AND J. A. SELLERS 3. Calculating the needed values of. From the above discussion, we can prove the congruences desired for all n after calculating the first M values of,forany M>5 α Cα+λ α. We calculate the necessary terms using recurrences. The recurrences needed for m are easily developed. Recurrences are suitable for calculating the values of m for small m. This, of course, is the historical approach to the calculation of partition function values. For example, this was the technique used by MacMahon to compute the first 200 values of pm [5, Table IV]. This same table was used by Ramanujan [8] in conjecturing several of the congruences in 1.5. We now prove a result from which the necessary recurrences follow. Theorem 3.1. nq n 1 n q n2 = pnq 2n q n Proof. From Jacobi s triple product identity [1, Theorem 2.8], we see that 1 n q n2 = q 2 ;q 2 q;q 2 2, q 2 ;q 2 5 q n2 = q;q 2 q4 ;q Also, since pnq n 1 =, 3.3 q;q it is clear that pnq 2n 1 = q2 ;q Then nq n 1 n q n2 q 2 ;q 4 = q;q 2 4 q4 ;q 4 q 2 ;q 2 2 = q;q 2 2 q4 ;q 4 2 q 2 ;q 2 q;q q 2 ;q 2 5 = q2 ;q 2 q;q 2 q4 ;q 4 2 = pnq 2n q n2. 3.5

7 COMPUTATIONAL PROOFS OF CONGRUENCES From this theorem, we have the following recurrences: 2k = pk+2 1 m+1 2k m 2 +2 m 1 m 1p k 2m 2, 2k+1 = 2 1 m+1 2k+1 m 2 +2 m 1 m 0p k 2mm Since pn satisfies pn = pn 1+pn 2 pn 5 pn 7+, where the values in question are the pentagonal numbers, the above recurrences can easily be implemented to calculate several values of. Using these recurrences, we have calculated the necessary 124, 216, 198 values of on a Linux PC with 768MB of RAM and a 600Mhz Pentium III processor. The calculations, all performed modulo 625, were completed in approximately 147 hours of computing time. With these calculations complete and the congruences checked modulo 625, Theorem 1.2 has been proven. 4. Closing remarks. While it would be nice to prove additional cases of 1.4 using this technique, it is clear that Cα grows too rapidly to make such an approach feasible. For example, the proof of the α = 5caseof1.4 would require the calculation of C5 = values of in the arithmetic progression 5 5 n+λ 5.Hence,we would have to calculate the first values of approximately. Certainly, a proof of Conjecture 1.1 via modular forms or generating function manipulations is still desired. This was originally requested in [9], and we renew that request here, given the new computational information that is now known about this partition function and the fact that Theorem 1.2 is proven. Acknowledgements. The authors gratefully acknowledge Dr David Gallagher and Mr Robert Schumacher of Cedarville University for their invaluable assistance in the computation of values of. The first author was partially supported by NSF VIGRE Grant # References [1] G. E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and Its Applications, vol. 2, Addison-Wesley, Massachusetts, [2], Generalized Frobenius partitions, Mem. Amer. Math. Soc , no. 301, iv+44. [3] D. Eichhorn and K. Ono, Congruences for partition functions, Analytic Number Theory, vol. 1 Allerton Park, IL, 1995, Progr. Math., vol. 138, Birkhäuser, Massachusetts, 1996, pp [4] B. Gordon and K. Hughes, Multiplicative properties of η-products. II, A Tribute to Emil Grosswald: Number Theory and Related Analysis, Contemp. Math., vol. 143, American Mathematical Society, Rhode Island, 1993, pp [5] G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proc. London Math. Soc , no. 2, [6] M. D. Hirschhorn and D. C. Hunt, A simple proof of the Ramanujan conjecture for powers of 5, J. Reine Angew. Math , [7] N. Koblitz, Introduction to Elliptic Curves and Modular Forms, 2nd ed., Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1993.

8 340 D. EICHHORN AND J. A. SELLERS [8] S. Ramanujan, Some properties of pn, the number of partitions of n, Proc. Cambridge Philos. Soc , [9] J. Sellers, Congruences involving F-partition functions, Int. J. Math. Math. Sci , no. 1, [10] J. Sturm, On the congruence of modular forms, Number Theory New York, , Lecture Notes in Math., vol. 1240, Springer, Berlin, 1987, pp Dennis Eichhorn: Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA address: James A. Sellers: Department of Mathematics, Penn State University, University Park, PA 16802, USA address:

9 Mathematical Problems in Engineering Special Issue on Time-Dependent Billiards Call for Papers This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration FA is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall. This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos both conservative and dissipative chaos. We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome. To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome. Authors should follow the Mathematical Problems in Engineering manuscript format described at Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at mts.hindawi.com/ according to the following timetable: Guest Editors Edson Denis Leonel, Department of Statistics, Applied Mathematics and Computing, Institute of Geosciences and Exact Sciences, State University of São Paulo at Rio Claro, Avenida 24A, 1515 Bela Vista, Rio Claro, SP, Brazil; edleonel@rc.unesp.br Alexander Loskutov, Physics Faculty, Moscow State University, Vorob evy Gory, Moscow , Russia; loskutov@chaos.phys.msu.ru Manuscript Due March 1, 2009 First Round of Reviews June 1, 2009 Publication Date September 1, 2009 Hindawi Publishing Corporation