Arithmetic properties arising from Ramanujan’s theta functions

We prove some interesting arithmetic properties of theta function identities that are analogous to q-series identities obtained by Michael D. Hirschhorn. In addition, we find infinite family of congruences modulo powers of 2 for representations of a non-negative integer n as △1+4△2 and △+k□

Some explicit values for ratios of theta-functions

In his notebooks [9], Ramanujan recorded several values of thetafunctions.B. C. Berndt and L-C. Zhang [6], Berndt and H. H. Chan[5] have proved all these evaluations. The main purpose of this paperis to establish several new explicit evaluations of ratios of thetafunctions. We also explicitly determine a(−2√2π),a(−2√2/9π),a(−√2π)and a(e−√2/9π), where a(q) is the Borweins cubic thetafunction. 2000 Mathematical Subject Classification: 33D15 , 33D20

Some explicit values for ratios of theta-functions.

In his notebooks [9], Ramanujan recorded several values of thetafunctions.B. C. Berndt and L-C. Zhang [6], Berndt and H. H. Chan[5] have proved all these evaluations. The main purpose of this paper is to establish several new explicit evaluations of ratios of thetafunctions. We also explicitly determine a(e−2√2π), a(e−2√2/9π), a(e −√2π)and a(e−√2/9π), where a(q) is the Borweins cubic theta-function

On 5-Regular Bipartitions with even Parts Distinct

In 2010, Andrews, Michael D. Hirschhorn and James A. Sellers considered the function ped(n), the number of partition of an integer n with even parts distinct (the odd parts are unrestricted). They obtained infinite families of congruences in the spirit of Ramanujan's congruences for the unrestricted partition function p(n). Let b(n) denote the number of 5-regular bipartitions of a positive integer n with even parts distinct (odd parts are unrestricted). In this paper, we establish many infinite families of congruences modulo powers of 2 for b(n). For example, ∑ n = 0 ∞ b 16 · 3 2 α · 5 2 β n + 14 · 3 2 α · 5 2 β + 1 q n = 8 f 2 3 f 5 3 ( mod 16 ) , where α , β ≥ 0

On 5-Regular Bipartitions with even Parts Distinct

In 2010, Andrews, Michael D. Hirschhorn and James A. Sellers considered the function ped(n), the number of partition of an integer n with even parts distinct (the odd parts are unrestricted). They obtained infinite families of congruences in the spirit of Ramanujan's congruences for the unrestricted partition function p(n). Let b(n) denote the number of 5-regular bipartitions of a positive integer n with even parts distinct (odd parts are unrestricted). In this paper, we establish many infinite families of congruences modulo powers of 2 for b(n). For example, ∑ n = 0 ∞ b 16 · 3 2 α · 5 2 β n + 14 · 3 2 α · 5 2 β + 1 q n = 8 f 2 3 f 5 3 ( mod 16 ) , where α , β ≥ 0

Congruences for Andrews' Singular Overpartitions

Many authors have found congruences and infinite families of congruences modulo 2, 3, 4, 18, and 36 for Andrews' defined combinatorial objects, called singular overpartitions, denoted by View the MathML source, which count the number of overpartitions of n in which no part is divisible by δ and only parts View the MathML source may be overlined. In this paper, we find congruences for View the MathML source modulo 4, 6, 12, 16, 18, and 72; infinite families of congruences modulo 12, 18, 48, and 72 for View the MathML source; and infinite families of congruences modulo 2 for View the MathML source, View the MathML source and View the MathML source. In addition, we find congruences for View the MathML source which represents the number of overpartitions where the parts are not multiples of 5

Arithmetic properties of partition k-tuples with odd parts distinct

Let pod−k(n) denote the number of partition k-tuples of n wherein odd parts are distinct(and even parts are unrestricted). We establish some interesting infinite families of congruences and internal congruences modulo 4, 16, and 5 for pod−2(n), pod−4(n),and pod−6(n), respectively. We also find Ramanujan-type congruences modulo 5 for pod−3(n) and densities of pod−2(n), pod−3(n), pod−4(n), and pod−6(n) modulo 4, 5,16, and 5, respectively

Congruences Modulo 2 for Certain Partition Functions

Let b3,5(n) denote the number of partitions of n into parts that are not multiples of 3 or 5. We establish several infinite families of congruences modulo 2 for b3,5(n). In the process, we also prove numerous parity results for broken 7-diamond partitions

Congruences for Overpartitions with Restricted Odd Differences

In recent work, Bringmann et al. used q-difference equations to compute a two-variable q-hypergeometric generating function for the number of overpartitions where (i) the difference between two successive parts may be odd only if the larger of the two is overlined, and (ii) if the smallest part is odd then it is overlined, given by t ¯ ( n ) . They also established the two-variable generating function for the same overpartitions where (i) consecutive parts differ by a multiple of ( k + 1 ) unless the larger of the two is overlined, and (ii) the smallest part is overlined unless it is divisible by k + 1 , enumerated by t ¯ ( k ) ( n ) . As an application they proved that t ¯ ( n ) = 0 ( mod 3 ) if n is not a square. In this paper, we extend the study of congruence properties of t ¯ ( n ) , and we prove congruences modulo 3 and 6 for t ¯ ( n ) , congruences modulo 2 and 4 for t ¯ ( 3 ) ( n ) and t ¯ ( 7 ) ( n ) , congruences modulo 4 and 5 for t ¯ ( 4 ) ( n ) , and congruences modulo 3, 6 and 12 for t ¯ ( 8 ) ( n )

Explicit evaluations of Ramanujan's remarkable product of theta-function.

On Page 259 of his second notebook [3], Ramanujan recorded many cubic modular equations of degree 2. In this paper we establish several cubic modular equations of degree 2 akin to those in Ramanujan’s work.As an application of our results, we also establish some new P − Q etafunction identities