The Pentagonal Number Theorem is one of Euler’s most profound discoveries. It is the following identity:

where n(3n − 1)/2 is called the n^th pentagonal numbers. The pentagonal numbers represent the number of distinct points which may be arranged to form superimposed regular pentagons such that the number of points on the sides of each respective pentagonal is the same. Bell’s article [2] is an excellent reference about the Pentagonal Number Theorem and its applications from the historical perspective. Andrews’s article [1] is devoted to a modern exposition of Euler’s original proof of the theorem.

Let ℕ = {1, 2, …} and ℕ₀ = {0, 1, 2, …}. The function p(n) is the number of integer partitions of n. The function σ(n) is the sum of all divisors of n. Euler applied the Pentagonal Number Theorem to study various properties of integer partitions and divisor sums. In particular, there is a convolution formula that connects the functions p(n) and σ(n):

Apart from this, there are recurrence relations for p(n) and σ(n) which are intimately related to the Pentagonal Number Theorem:

The readers are invited to read the survey article by Osler et. al. [4] for a readable account of the connections between functions p(n) and σ(n).

One may wonder if there are recurrence relations and convolution formulas for (restricted) integer partitions and divisor sums in terms of other polygonal numbers (e.g. triangular numbers, heptagonal numbers …). The goal of this article is to give positive answers to this question based on a generalization of the Pentagonal Number Theorem.

Unless mentioned otherwise, throughout the paper, all equations in the variable q which involve infinite sums and infinite products hold true if |q| < 1.

2.1. Main Theorem and Some Corollaries

Let g > 3 and n ∈ ℤ. We denote the generalized n^thg-gonal number [6, p.40] by

For n ∈ ℕ, P_g,n represents the number of points which may be arranged to form regular g-gons such that the number of points on the sides of each respective g-gon is the same as n.

Let (a; q)_n be the q-Pochhammer symbol for n ≥ 1. That is,

Considered as a formal power series in q, the definition of q-Pochhammer symbol can be extended to an infinite product. That is,

We note that (q; q)_∞ is the Euler’s function.

We state the following theorem.

Definition 2.2.1

The function (p_2,5 + p_3,5)(n) is the number of partitions of n such that each part is either congruent to 2 modulo 5 or 3 modulo 5. We extend the domain of (p_2,5 + p_3,5)(n) to ℤ by setting (p_2,5 + p_3,5)(x) = 0 if x ∉ ℕ₀.

Theorem 2.2.2

Example 2.2.3

Let n = 8. Then we have 8 = 2 + 2 + 2 + 2 = 3 + 3 + 2. So, (p_2,5 + p_3,5)(8) = 3. The generalized heptagonal numbers are P_7,1 = 1, Q_7,1 = 4, P_7,2 = 7… and hence

Definition 2.2.4

The function (p_1,5 + p_4,5)(n) is the number of partitions of n such that each part is either congruent to 1 modulo 5 or 4 modulo 5. We extend the domain of (p_1,5 + p_4,5)(n) to ℤ by setting (p_1,5 + p_4,5)(x) = 0 if x ∉ ℕ₀.

Theorem 2.2.5

where

Example 2.2.6

Let n = 9. The generalized heptagonal numbers are P_7,1 = 1, Q_7,1 = 4, P_7,2 = 7…. By Theorem 2.2.5, we get

It can be easily verified since

Definition 2.2.7

The function q(n) is the number of partitions of n such that each part is distinct. We extend the domain of q(n) to ℚ by setting q(x) = 0 if x ∉ ℕ₀.

We obtain two recurrence relations for q(n).

Theorem 2.2.8

where

Theorem 2.2.9

where

Example 2.2.10

Let n = 15. We note that 15 = Δ₅. The generalized pentagonal numbers are

By Theorem 2.2.8,

Alternatively, by Theorem 2.2.9 and the fact that 2(15) = 30 ≠ Δ_m for any m ∈ ℕ.

Definition 2.2.11

Let n, r ∈ ℕ₀. Let m ∈ ℕ. The function p_r,m(n) is defined to be the number of partitions of n such that each part is congruent to r modulo m. We extend the domain of p_r,m(n) to ℤ by setting p_r,m(x) = 0 if x ∉ ℕ₀.

Definition 2.2.12

Let n ∈ ℤ and m ∈ ℕ. The restricted integer partition pm′(n) is defined by

where pm′(0):=1 and pm′(n)=0 for n ∉ ℕ₀.

In particular, if m = 3, then pm′(n)=p3′(n)=p(n). The generating function of pm′(n) is as follows:

It is convenient to introduce the number e_g,n given by

Theorem 2.1.1 can be rewritten as

We obtain the following recurrence relation for the restricted partition pm′(n) if m ≥ 3.

Theorem 2.2.13

Let m ≥ 3 and n ∈ ℕ.

Remark 2.2.14

In the case m = 3, Theorem 2.2.13 is reduced to the well-known recurrence relation for the integer partition p(n).

Definition 2.2.15

The function pe,m′(n) is the number of partitions of n such that each partition has an even number of distinct parts, and each part is congruent to 0 modulo m, 1 modulo m or m − 1 modulo m.

Definition 2.2.16

The function po,m′(n) is the number of partitions of n such that each partition has an odd number of distinct parts, and each part is congruent to 0 modulo m, 1 modulo m or m − 1 modulo m.

Theorem 2.2.17

Let n ∈ ℕ and m ≥ 3.

Remark 2.2.18

In the case m = 3, Theorem 2.2.17 is reduced to the interesting observation made by Legendre on the Pentagonal Number Theorem [1, p.2].

Definition 2.3.1

Let r ∈ ℕ₀. Let n, m ∈ ℕ. The function σ_r,m(n) is defined by

Definition 2.3.2

Let n, m ∈ ℕ. The restricted divisor sum σm′(n) is defined by

The restricted Lambert series for σ_r,m(n) is

The restricted Lambert series for σm′(n) is

The following theorems connect pm′(n), σm′(n) and the generalized n^th-gonal numbers.

Theorem 2.3.3

Let m ≥ 3 and n ∈ ℕ.

Theorem 2.3.4

Let m ≥ 3 and n ∈ ℕ.

Theorem 2.3.5

Let m ≥ 3 and n ∈ ℕ.

Remark 2.3.6

In the case m = 3, Theorem 2.3.5 is identical to (1.1).

3.1. Preliminaries

Let (x₁, x₂, …) be a sequence of real numbers. The partial exponential Bell polynomials are polynomials given by

where π(n, k) is the positive integer sequence (j₁, j₂, j_n−k+1) satisfies the following equations:

For n ≥ 1, the n^th-complete Bell polynomial B_n(x₁, …, x_n) is the following:

The complete exponential Bell polynomials can also be defined by power series expansion as follows:

where B₀ ≡ 1. Alternatively, the complete Bell polyomials can be recursively defined by

One interesting property of the Bell polynomials is that there exists an inversion formula in the following sense. If we define

then

For detailed properties of such inverse formulas, see the paper written by Chaou et. al [5]. Bell polynomials were first introduced by Bell [2]. The books written by Comtet [7] and Riordan [10] serve as excellent references for the numerous applications of Bell polynomials in combinatorics. Recently, there has been extensive research in finding identities on (partial/complete) Bell polynomials. The paper written by W. Wang and T. Wang [12] provides many interesting identities for partial Bell polynomials. Bouroubi and Benyahia-Tani [3] and the author [8] proved some new identities for complete Bell polynomials based on Ramanujan’s congruences.

3.2. Identities for Complete Bell Polynomials and Some Corollaries

We recall the notations σm′(n),pm′(n) and e_g,n used in Section 2.2 and Section 2.3.

The author is grateful to the editor and the referees for carefully reading the paper and pointing out some mistakes in the first draft of it. Their comments were helpful to improve the quality of the article. The author is supported by UAEU Startup Grant 2016 (G00002235) from United Arab Emirates University.