The Ramanujan Journal

This paper provides elementary proofs for several types of congruences involving multipartitions and self-convolutions of the divisor function. Our computations use differential algebra and Gröbner bases, and employ a couple of MAPLE programs available as ancillary files on arXiv. The first results of the paper are Ramanujan-type congruences of the form $$p^{*k}(qn+r) \equiv _q 0$$ and $$\sigma ^{*k}(qn+r) \equiv _q 0$$ , where p(n) and $$\sigma (n)$$ are the partition and divisor functions, $$q > 3$$ is prime, $$\equiv _q$$ denotes congruence modulo q, and $$^{*k}$$ denotes kth-order self-convolution. We prove all the valid congruences of this form for $$q \in \{5, 7, 11\}$$ , including the three famous partition congruences of Ramanujan, and an exceptional one for $$q = 17$$ . All such multipartition congruences have already been settled in principle (up to a numerical verification) due to Eichhorn and Ono, using modular forms. On the other hand, the majority of the divisor function congruences are new results. We then proceed to search for more general congruences modulo small primes, concerning linear combinations of $$\sigma ^{*k}(qn+r)$$ for different values of k, as well as weighted convolutions of p(n) and $$\sigma (n)$$ with polynomial weights. The paper ends with a few corollaries and extensions for the divisor function congruences, including proofs for three conjectures of N.C. Bonciocat.

Let k∈{10,15,20}, and let b k (n) denote the number k-regular partitions of n. We prove for half of all primes p and any t≥1 that there exist p−1 arithmetic progressions modulo p 2t such that b k (n) is a multiple of 5 for each n in one of these progressions.

We find new partition identities arising from Ramanujan’s formulas of multipliers. Several of the identities are for overpartitions, overpartition pairs, and $$\ell $$ -regular partitions.

In recent work, Miezaki introduced the notion of a spherical T-design in $$\mathbb {R}^2$$ , where T is a potentially infinite set. As an example, he offered the $$\mathbb {Z}^2$$ -lattice points with fixed integer norm (a.k.a. shells). These shells are maximal spherical T-designs, where $$T=\mathbb {Z}^+\setminus 4\mathbb {Z}^+$$ . We generalize the notion of a spherical T-design to special ellipses, and extend Miezaki’s work to the norm form shells for rings of integers of imaginary quadratic fields with class number 1.

This paper gives a solution, without the use of the three-term recurrence relation, of the problem posed in Ismail (Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, Cambridge, 2005) (Problem 24.8.2, p. 658): that the hypergeometric representation of the general Pollaczek polynomials is a polynomial in cos(θ) of degree n. Chu solved in (Ramanujan J. 13(1–3): 221–225, 2007) the problem in a particular case. We use elementary properties of functions of complex variables and Pfaff’s transformation on hypergeometric 2 F 1-series.

We construct a class of companion elliptic functions associated with even Dirichlet characters. Using the well-known properties of the classical Weierstrass elliptic function $$\wp (z|\tau )$$ as a blueprint, we will derive their representations in terms of q-series and partial fractions. We also explore the significance of the coefficients of their power series expansions and establish the modular properties under the actions of the arithmetic groups $$\Gamma _0(N)$$ and $$\Gamma _1(N)$$ .

We find several new congruences for $$\ell $$ -regular partitions for $$\ell \in \{5,6,7,49\}$$ and also find alternative proofs of the congruences for 10- and 20-regular partitions which were proved earlier by Carlson and Webb (Ramanujan J 33:329–337, 2014) by using the theory of modular forms. We use certain p-dissections of $$(q;q)_{\infty }$$ , $$\psi (q)$$ , $$(q;q)_{\infty }^3$$ and $$\psi (q^2)(q;q)_{\infty }^2$$ .

We show how a simplicial complex arising from the WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) equations of string theory is the Whitehouse complex. Using discrete Morse theory, we give an elementary proof that the Whitehouse complex Δ n is homotopy equivalent to a wedge of (n−2)! spheres of dimension n−4. We also verify the Cohen-Macaulay property. Additionally, recurrences are given for the face enumeration of the complex and the Hilbert series of the associated pre-WDVV ring.

In 2003, Maróti showed that one could use the machinery of ℓ-cores and ℓ-quotients of partitions to establish lower bounds for p(n), the number of partitions of n. In this paper we explore these ideas in the case ℓ=2, using them to give a largely combinatorial proof of an effective upper bound on p(n), and to prove asymptotic formulae for the number of self-conjugate partitions, and the number of partitions with distinct parts. In a further application we give a combinatorial proof of an identity originally due to Gauss.