Andrews, Lewis and Lovejoy introduced the partition function P D(n) as the number of partitions of n with designated summands, where we assume that among parts with equal size, exactly one is designated. They proved that P D(3n + 2) is divisible by 3. We obtain a Ramanujan type identity for the generating function of P D(3n + 2) which implies the congruence of Andrews, Lewis and Lovejoy. For P D(3n), Andrews, Lewis and Lovejoy showed that the generating function can be expressed as an infinite product of powers of (1 − q 2n+1 ) times a function F (q 2 ). We find an explicit formula for F (q 2 ), which leads to a formula for the generating function of P D(3n). We also obtain a formula for the generating function of P D(3n + 1). Our proofs rely on Chan's identity on Ramanujan's cubic continued fraction and some identities on cubic theta functions. By introducing a rank for the partitions with designed summands, we give a combinatorial interpretation of the congruence of Andrews, Lewis and Lovejoy.

We use both Abel's lemma on summation by parts and Zeilberger's algorithm to find recurrence relations for definite summations. The role of Abel's lemma can be extended to the case of linear difference operators with polynomial coefficients. This approach can be used to verify and discover identities involving harmonic numbers and derangement numbers. As examples, we use the Abel-Zeilberger algorithm to prove the Paule-Schneider identities, the Apéry-Schmidt-Strehl identity, Calkin's identity and some identities involving Fibonacci numbers.

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Andrews, Brietzke, Rødseth and Sellers proved an infinite family of congruences on the number of the restricted m-ary partitions when m is a prime. In this note, we show that these congruences hold for arbitrary positive integer m and thus confirm the conjecture of Andrews, et al.

Let b ℓ (n) be the number of ℓ-regular partitions of n. Recently, Hou et al established several infinite families of congruences for b ℓ (n) modulo m, where (ℓ, m) = (3, 3), (6, 3), (5, 5), (10, 5) and (7, 7). In this paper, by the vanishing property given by Hou et al, we show an infinite family of congruence for b 11 (n) modulo 11. Moreover, for ℓ = 3, 13 and 25, we obtain three infinite families of congruences for b ℓ (n) modulo 3, 5 and 13 by the theory of Hecke eigenforms.