Determinants with Bernoulli polynomials and the restricted partition function

TL;DR

This paper explores determinants involving Bernoulli polynomials and their relationship with the restricted partition function, providing new insights into counting solutions to linear Diophantine equations.

Contribution

It introduces two natural determinants with Bernoulli polynomials and connects them to the restricted partition function, offering novel analytical tools.

Findings

Established connections between Bernoulli polynomial determinants and partition functions

Derived formulas linking determinants to counting solutions of linear equations

Enhanced understanding of the structure of restricted partition functions

Abstract

Let $r\geq 1$ be an integer, $\mathbf a=(a_1,\ldots,a_r)$ a vector of positive integers and let $D\geq 1$ be a common multiple of $a_1,\ldots,a_r$. We study two natural determinants of order $rD$ with Bernoulli polynomials and we present connections with the restricted partition function $p_{\mathbf a}(n):=$ the number of integer solutions $(x_1,\dots,x_r)$ to $\sum_{j=1}^r a_jx_j=n$ with $x_1\geq 0, \ldots, x_r\geq 0$.

Equations

$$F_{r,D}(x_1,\dots ,x_D):=\left|\begin{array}{ccccccc}\frac{B_1(x_1)}{1}& \cdots & \frac{B_1(x_D)}{1}& \cdots & \frac{B_r(x_1)}{r}& \cdots & \frac{B_r(x_D)}{r}\\ \frac{B_2(x_1)}{2}& \cdots & \frac{B_2(x_D)}{2}& \cdots & \frac{B_{r+1}(x_1)}{r+1}& \cdots & \frac{B_{r+1}(x_D)}{r+1}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ \frac{B_{rD}(x_1)}{rD}& \cdots & \frac{B_{rD}(x_D)}{rD}& \cdots & \frac{B_{rD+r-1}(x_1)}{rD+r-1}& \cdots & \frac{B_{rD+r-1}(x_D)}{rD+r-1}\end{array}\right|\in \mathbb{Q}[x_1,\dots ,x_D],$$

$${\Delta }_{r,D}=(-1{)}^{\frac{rD(rD+r)}{2}}{D}^{\frac{rD(rD+r-2)}{2}}\cdot F_{r,D}(\frac{D-1}{D},\dots ,\frac{1}{D},0).$$

$$F_{1,D}(x_1,\dots ,x_D)=\frac{1}{D!}\prod _{1\le i<j\le D}(x_j-x_i)\sum _{t=0}^D(-1{)}^t\frac{E_{D,D-t}(x_1,\dots ,x_D)}{t+1},$$

$$F_{r,D}(x_1,\dots ,x_D)=\left(\prod _{1\le i<j\le D}(x_j-x_i{)}^r\right)G_{r,D}(x_1,\dots ,x_D),$$

$$\sum _{v=0}^{D-1}(-D{)}^m{d}_{\mathbf{a},m}(v)=\frac{(-1{)}^{r-1}{D}^{m+1}}{m!(r-1-m!)}{B}_{r-1-m}(\mathbf{a}),(\forall )0\le m\le r-1.$$

$$\sum _{m=0}^{r-1}\sum _{v=1}^D{d}_{\mathbf{a},m}(v)D^{n+m}\frac{B_{n+m+1}(\frac{v}{D})}{n+m+1}=\frac{(-1{)}^{r-1}n!}{(n+r)!}{B}_{r+n}(\mathbf{a})-{\delta }_{0n},(\forall )0\le n\le rD-r-1,$$

$${\bar{F}}_{r,D}(x_1,\dots ,x_D):=\left|\begin{array}{ccccccc}1& \cdots & 1& \cdots & 0& \cdots & 0\\ ⋮& \cdots & ⋮& \cdots & ⋮& \cdots & ⋮\\ 0& \cdots & 0& \cdots & 1& \cdots & 1\\ B_1(x_1)& \cdots & B_1(x_D)& \cdots & \frac{B_r(x_1)}{r}& \cdots & \frac{B_r(x_D)}{r}\\ ⋮& ⋮& ⋮& ⋮& ⋮& & \\ \frac{B_{rD-r}(x_1)}{rD-r}& \cdots & \frac{B_{rD-r}(x_D)}{rD-r}& \cdots & \frac{B_{rD-1}(x_1)}{rD-1}& \cdots & \frac{B_{rD-1}(x_D)}{rD-1}\end{array}\right|.$$

$${\bar{F}}_{r,D}(x_1,\dots ,x_D)=(-1{)}^{(D+1)\left(\genfrac{}{}{0ex}{}{r}{2}\right)}\left(\prod _{1\le i<j\le D}(x_j-x_i{)}^r\right)G_{r,D}(x_1,\dots ,x_D),$$

$$(1)F_{1,D}(x_1,\dots ,x_D)=\frac{1}{(D-1)!}\prod _{1\le i<j\le D}(x_j-x_i),(2){\bar{\Delta }}_{1,D}=\frac{1!2!\cdots (D-2)!}{(-D{)}^D}.$$

$${\bar{F}}_{r,2}(x_1,x_2)=(-1{)}^{\left(\genfrac{}{}{0ex}{}{r}{2}\right)}\frac{[1!2!\cdots (r-1)!{]}^3}{r!(r+1)!\cdots (2r-1)!}(x_2-x_1{)}^r\prod _{j=0}^{r-1}((x_2-x_1{)}^2-j^2{)}^{r-j},(\forall )r\ge 1,$$

$$p_{\mathbf{a}}(n):=\#\{(x_1,\dots ,x_r)\in {\mathbb{N}}^r:\sum _{i=1}^r{a}_i{x}_i=n\},(\forall )n\ge 0.$$

$$p_{\mathbf{a}}(n)=d_{\mathbf{a},r-1}(n)n^{r-1}+\cdots +d_{\mathbf{a},1}(n)n+d_{\mathbf{a},0}(n),(\forall )n\ge 0,$$

$${\zeta }_{\mathbf{a}}(s,w):=\sum _{n=0}^{\infty }\frac{p_{\mathbf{a}}(n)}{(n+w{)}^s},\mathrm{Re}s>r,$$

$${\zeta }_{\mathbf{a}}(s):=\underset{w\searrow 0}{\lim }({\zeta }_{\mathbf{a}}(s,w)-w^{-s}).$$

$${\zeta }_{\mathbf{a}}(s)=\frac{1}{D^s}\sum _{m=0}^{r-1}\sum _{v=1}^D{d}_{\mathbf{a},m}(v)D^m\zeta (s-m,\frac{v}{D}),$$

$$\frac{z}{e^z-1}=\sum _{j=0}^{\infty }{B}_j\frac{z^j}{j!},$$

$$\frac{z{e}^{xz}}{(e^z-1)}=\sum _{n=0}^{\infty }{B}_n(x)\frac{z^n}{n!}.$$

$$B_n(x)=\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)B_{n-k}{x}^k.$$

$$\frac{z^r{e}^{xz}}{(e^{a_{1}z}-1)\cdots (e^{a_{r}z}-1)}=\sum _{j=0}^{\infty }{B}_j(x;\mathbf{a})\frac{z^j}{j!}.$$

$$B_j(\mathbf{a}):=B_j(0;\mathbf{a})=\sum _{i_1+\cdots +i_r=j}\left(\genfrac{}{}{0ex}{}{j}{i_1,\dots ,i_r}\right)B_{i_1}\cdots B_{i_r}{a}_1^{i_1-1}\cdots a_r^{i_r-1}.$$

$$\sum _{m=0}^{r-1}\sum _{v=1}^D{d}_{\mathbf{a},m}(v)D^{n+m}\frac{B_{n+m+1}(\frac{v}{D})}{n+m+1}=\frac{(-1{)}^{r-1}n!}{(n+r)!}{B}_{r+n}(\mathbf{a})-{\delta }_{0n},(\forall )n\in \mathbb{N},$$

$${\Delta }_{r,D}:=\left|\begin{array}{ccccccc}\frac{B_1(\frac{1}{D})}{1}& \cdots & \frac{B_1(1)}{1}& \cdots & D^{r-1}\frac{B_r(\frac{1}{D})}{r}& \cdots & D^{r-1}\frac{B_r(1)}{r}\\ D\frac{B_2(\frac{1}{D})}{2}& \cdots & D\frac{B_2(1)}{2}& \cdots & D^r\frac{B_{r+1}(\frac{1}{D})}{r+1}& \cdots & D^r\frac{B_{r+1}(1)}{r+1}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ D^{rD-1}\frac{B_{rD}(\frac{1}{D})}{rD}& \cdots & D^{rD-1}\frac{B_{rD}(1)}{rD}& \cdots & D^{rD+r-2}\frac{B_{rD+r-1}(\frac{1}{D})}{rD+r-1}& \cdots & D^{rD+r-2}\frac{B_{rD+r-1}(1)}{rD+r-1}\end{array}\right|$$

$$B_n(1-x)=(-1{)}^n{B}_n(x)\text{ for all }n\ge 0,$$

$${\Delta }_{r,D}=(-1{)}^{\frac{rD(rD+r)}{2}}{D}^{\frac{rD(rD+r-2)}{2}}\left|\begin{array}{ccccccc}\frac{B_1(\frac{D-1}{D})}{1}& \cdots & \frac{B_1(0)}{1}& \cdots & \frac{B_r(\frac{D-1}{D})}{r}& \cdots & \frac{B_r(0)}{r}\\ \frac{B_2(\frac{D-1}{D})}{2}& \cdots & \frac{B_2(0)}{2}& \cdots & \frac{B_{r+1}(\frac{D-1}{D})}{r+1}& \cdots & \frac{B_{r+1}(0)}{r+1}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ \frac{B_{rD}(\frac{D-1}{D})}{rD}& \cdots & \frac{B_{rD}(0)}{rD}& \cdots & \frac{B_{rD+r-1}(\frac{D-1}{D})}{rD+r-1}& \cdots & \frac{B_{rD+r-1}(0)}{rD+r-1}\end{array}\right|.$$

$$d_{\mathbf{a},m}(v)=\frac{{\Delta }_{r,D}^{m,v}}{{\Delta }_{r,D}},(\forall )1\le v\le D,0\le m\le r-1,$$

$$p_{\mathbf{a}}(n)=\frac{1}{{\Delta }_{r,D}}\sum _{m=0}^{r-1}{\Delta }_{r,D}^{m,v}{n}^m,(\forall )n\in \mathbb{N}.$$

$$F_{r,D}(x_1,\dots ,x_D):=\left|\begin{array}{ccccccc}\frac{B_1(x_1)}{1}& \cdots & \frac{B_1(x_D)}{1}& \cdots & \frac{B_r(x_1)}{r}& \cdots & \frac{B_r(x_D)}{r}\\ \frac{B_2(x_1)}{2}& \cdots & \frac{B_2(x_D)}{2}& \cdots & \frac{B_{r+1}(x_1)}{r+1}& \cdots & \frac{B_{r+1}(x_D)}{r+1}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ \frac{B_{rD}(x_1)}{rD}& \cdots & \frac{B_{rD}(x_D)}{rD}& \cdots & \frac{B_{rD+r-1}(x_1)}{rD+r-1}& \cdots & \frac{B_{rD+r-1}(x_D)}{rD+r-1}\end{array}\right|\in \mathbb{Q}[x_1,\dots ,x_D].$$

$${\Delta }_{r,D}=(-1{)}^{\frac{rD(rD+r)}{2}}{D}^{\frac{rD(rD+r-2)}{2}}\cdot F_{r,D}(\frac{D-1}{D},\dots ,\frac{1}{D},0).$$

$$\Delta :=\left|\begin{array}{cccc}1& \frac{1}{2}& \cdots & \frac{1}{r}\\ \frac{1}{2}& \frac{1}{3}& \cdots & \frac{1}{r+1}\\ ⋮& ⋮& \ddots & ⋮\\ \frac{1}{r}& \frac{1}{r+1}& \cdots & \frac{1}{2r-1}\end{array}\right|=\frac{[1!2!\cdots (r-1)!{]}^3}{r!(r+1)!\cdots (2r-1)!}.$$

$${\Delta }_{\ell }:=\left|\begin{array}{cccc}\frac{r!}{\ell }& \frac{r!}{\ell +1}& \cdots & \frac{r!}{r}\\ \frac{(r+1)!}{\ell +1}& \frac{(r+1)!}{\ell +2}& \cdots & \frac{(r+1)!}{r+1}\\ ⋮& ⋮& \ddots & ⋮\\ \frac{(2r-\ell )!}{r}& \frac{(2r-\ell )!}{r+1}& \cdots & \frac{(2r-\ell )!}{2r-\ell }\end{array}\right|.$$

Full text

Keywords: restricted partition function, Bernoulli polynomial, Bernoulli Barnes numbers.

2010 MSC: Primary 11P81 ; Secondary 11B68, 11P82

1 Introduction

Let $\mathbf{a}:=(a_{1},a_{2},\ldots,a_{r})$ be a sequence of positive integers, $r\geq 1$. The restricted partition function associated to $\mathbf{a}$ is $p_{\mathbf{a}}:\mathbb{N}\rightarrow\mathbb{N}$, $p_{\mathbf{a}}(n):=$ the number of integer solutions $(x_{1},\ldots,x_{r})$ of $\sum_{i=1}^{r}a_{i}x_{i}=n$ with $x_{i}\geq 0$. Let $D$ be a common multiple of $a_{1},\ldots,a_{r}$. The restricted partition function $p_{\mathbf{a}}(n)$ was studied extensively in the literature, starting with the works of Sylvester [13] and Bell [3]. Popoviciu [10] gave a precise formula for $r=2$. Recently, Bayad and Beck [2, Theorem 3.1] proved an explicit expression of $p_{\mathbf{a}}(n)$ in terms of Bernoulli-Barnes polynomials and the Fourier Dedekind sums, in the case that $a_{1},\ldots,a_{r}$ are are pairwise coprime.

Let $D$ be a common multiple of $a_{1},\ldots,a_{r}$. In [7], we reduced the computation of $p_{\mathbf{a}}(n)$ to solving the linear congruence $a_{1}j_{1}+\cdots+a_{r}j_{r}\equiv n(\bmod\;D)$ in the range $0\leq j_{1}\leq\frac{D}{a_{1}}-1,\ldots,0\leq j_{r}\leq\frac{D}{a_{r}}-1$. In [8], we proved that if a determinant $\Delta_{r,D}$, see (2.5), which depends only on $r$ and $D$, with entries consisting in values of Bernoulli polynomials is nonzero, then $p_{\mathbf{a}}(n)$ can be computed in terms of values of Bernoulli polynomials and Bernoulli Barnes numbers. In the second section, we outline several construction and results from [8]. In the third section, we study the polynomial

[TABLE]

which is related to $\Delta_{r,D}$ by the identity

[TABLE]

In Theorem $3.4$ we prove that

[TABLE]

where $E_{D,0}(x_{1},\ldots,x_{D})=1$, $E_{D,1}(x_{1},\ldots,x_{D})=x_{1}+\cdots+x_{D}$ etc. are the elementary symmetric polynomials. In Proposition $3.6$, we prove that

[TABLE]

where $G_{r,D}(x_{1},\ldots,x_{D})$ is a symmetric polynomial, hence $\Delta_{r,D}\neq 0$ iff $G_{r,D}(\frac{D-1}{D},\ldots,0)\neq 0$.

In the last section, we propose another approach to the initial problem, studied in [8], of computing $p_{\mathbf{a}}(n)$ in terms of values of Bernoulli polynomials and Bernoulli Barnes numbers. In formula $(4.3)$ we show that

[TABLE]

Seeing $d_{\mathbf{a},m}(v)$’s as indeterminates and considering also the identities

[TABLE]

we obtain a system of $rD$ linear equations with a determinant $\bar{\Delta}_{r,D}$. In Remark $4.1$ we note that if $\bar{\Delta}_{r,D}\neq 0$, then $d_{\mathbf{a},m}(v)$, $0\leq m\leq r-1$, $1\leq v\leq D$, are the unique solutions of the above system. We consider the polynomial $\bar{F}_{r,D}\in\mathbb{Q}[x_{1},\ldots,x_{D}]$ defined by

[TABLE]

We have that $\bar{\Delta}_{r,D}=(-D)^{D\binom{r}{2}+\binom{rD-r}{2}}\bar{F}_{r,D}(\frac{D-1}{D},\ldots,\frac{1}{D},0)$. In formula $(4.9)$ we show that

[TABLE]

where $G_{r,D}$ is a symmetric polynomial with $\deg G_{r,D}\leq r\binom{rD-r}{2}+D\binom{r}{2}-r\binom{D}{2}$.

Using the methods of Olson [9], in Proposition $4.2$ we prove that for any $D\geq 1$ we have

[TABLE]

By our computer experiments in Singular [6], we expect that the following formula holds

[TABLE]

some justifications being noted in Remark $4.3$. Also, we propose a formula for $\bar{F}_{2,D}$, see Conjecture $4.5$, but we are unable to “guess” a formula for $\bar{F}_{r,D}$ in general.

2 Preliminaries

Let $\mathbf{a}:=(a_{1},a_{2},\ldots,a_{r})$ be a sequence of positive integers, $r\geq 1$. The restricted partition function associated to $\mathbf{a}$ is $p_{\mathbf{a}}:\mathbb{N}\rightarrow\mathbb{N}$,

[TABLE]

Let $D$ be a common multiple of $a_{1},\ldots,a_{r}$. Bell [3] has proved that $p_{\mathbf{a}}(n)$ is a quasi-polynomial of degree $r-1$, with the period $D$, i.e.

[TABLE]

where $d_{\mathbf{a},m}(n+D)=d_{\mathbf{a},m}(n)$, $(\forall)0\leq m\leq r-1,n\geq 0$, and $d_{\mathbf{a},r-1}(n)$ is not identically zero. The Barnes zeta function associated to $\mathbf{a}$ and $w>0$ is

[TABLE]

see [1] and [12] for further details. It is well known that $\zeta_{\mathbf{a}}(s,w)$ is meromorphic on $\mathbb{C}$ with poles at most in the set $\{1.\ldots,r\}$. We consider the function

[TABLE]

In [7, Lemma 1.6] we proved that

[TABLE]

where $\zeta(s,w):=\sum_{n=0}^{\infty}\frac{1}{(n+w)^{s}},\;\operatorname{Re}s>1,$ is the Hurwitz zeta function. The Bernoulli numbers $B_{j}$ are defined by

[TABLE]

$B_{0}=1$, $B_{1}=-\frac{1}{2}$, $B_{2}=\frac{1}{6}$, $B_{4}=-\frac{1}{30}$ and $B_{n}=0$ if $n$ is odd and greater than $1$. The Bernoulli polynomials are defined by

[TABLE]

They are related with the Bernoulli numbers by

[TABLE]

The Bernoulli-Barnes polynomials are defined by

[TABLE]

The Bernoulli-Barnes numbers are defined by

[TABLE]

In [8, Formula (2.9)] we proved that

[TABLE]

where $\delta_{0n}=\begin{cases}1,&n=0,\\ 0,&n\geq 1\end{cases}$ is the Kronecker symbol. Given values $0\leq n\leq rD-1$ in $(1.9)$ and seeing $d_{\mathbf{a},m}(v)$’s as indeterminates, we obtain a system of linear equations with the determinant

[TABLE]

Using basic properties of determinants and the fact that

[TABLE]

it follows that

[TABLE]

Proposition 2.1**.**

(See [8, Proposition 2.1] and [8, Corollary 2.2])

With the above notations, if $\Delta_{r,D}\neq 0$, then

[TABLE]

where $\Delta_{r,D}^{m,v}$ is the determinant obtained from $\Delta_{r,D}$, as defined in $(\ref{pista})$, by replacing the $(mD+v)$-th column with the column $(\frac{(-1)^{r-1}n!}{(n+r)!}B_{n+r}(\mathbf{a})-\delta_{n0})_{0\leq n\leq rD-1}$. Consequently,

[TABLE]

Proof.

The first part follows from the Cramer rule applied to the system (2.4). The second part is a consequence of the first part and (2.1). ∎

Remark 2.2**.**

In [8] it was conjectured that $\Delta_{r,D}\neq 0$ for any $r,D\geq 1$. An affirmative answer was given in the case $r=1$, $r=2$ and $D=1$. In the general case, an equivalent form was given in [8, Theorem 2.3], which reduced the problem to show that a $r\times r$ determinant is non zero. In the next section we tackle this problem from another point of vue, by studying a polynomial $F_{r,D}$ is $D$ indeterminates with the property that $\Delta_{r,D}=F_{r,D}(\frac{D-1}{D},\ldots,\frac{1}{D},0)$.***

3 Determinants with Bernoulli polynomials

Let $r,D\geq 1$ be two integers. We consider the polynomial

[TABLE]

According to (2.6) and (3.1), using the notations from the previous section, we have that

[TABLE]

Lemma 3.1**.**

For any $r\geq 1$ we have that

[TABLE]

Proof.

We let

[TABLE]

Note that $\Delta=r!(r+1)!\cdots(2r-1)!\Delta_{1}$. We have $\Delta_{r}=(r-1)!$. For $1\leq\ell<r$, we have

[TABLE]

Multiplying the first line accordingly and adding to the next lines in order to obtain zeroes on the last column, it follows that

[TABLE]

hence the induction step is complete. ∎

Proposition 3.2**.**

We have that

[TABLE]

Proof.

We have $B_{n}(x)=x^{n}+$ terms of lower order, hence the result follows from Lemma $3.1$. ∎

Proposition 3.3**.**

For $r=1$ and $D\geq 1$ the following hold:

1. (1)

There exists a symetric polynomial $G_{1,D}(x_{1},\ldots,x_{D})$ of degree $D$ such that

[TABLE] 2. (2)

$G_{1,D}(x_{1},\ldots,x_{D})=\frac{1}{D!}x_{1}x_{2}\cdots x_{D}+$* terms of lower degree.* 3. (3)

$G_{1,D}(0,\ldots,0)=\frac{(-1)^{D}}{(D+1)!}$.

Proof.

(1) From (3.1) it follows that

[TABLE]

Moreover, for any permutation $\sigma\in S_{D}$, we have that

[TABLE]

Since

[TABLE]

from $(\ref{1D})$ and $(\ref{coocoo})$ it follows that

[TABLE]

where $G_{1,D}\in\mathbb{Q}[x_{1},\ldots,x_{D}]$ is a symmetrical polynomial of degree $D$.

(2) The homogeneous component of highest degree of $F_{1,D}$ is

[TABLE]

hence $G_{1,D}(x_{1},\ldots,x_{D})=\frac{1}{D!}x_{1}\cdots x_{D}+$ terms of lower order.

(3) For any integers $j\geq 0$ and $1\leq n\leq D$, we let

[TABLE]

i.e. $L_{1}(x_{1},\ldots,x_{n})=x_{1}+\cdots+x_{n}$, $L_{2}(x_{1},\ldots,x_{n})=x_{1}^{2}+\cdots+x_{n}^{2}+x_{1}x_{2}+\cdots+x_{n-1}x_{n}$, etc. It is easy to check that

[TABLE]

We let

[TABLE]

Inductively, for $1<j\leq k\leq D$ and $\ell\geq 1$, we define

[TABLE]

We prove by induction on $j\geq 1$ that

[TABLE]

Indeed, since $B_{\ell}(x)=\sum_{t=0}^{\ell}\binom{\ell}{t}B_{\ell-t}x^{t}$, it follows that (3.8) holds for $j=1$. Now, assume that $j\geq 2$. From the induction hypothesis, (3.7), (3.6) and (3.8) it follows that

[TABLE]

[TABLE]

hence the induction step is complete. Using standard properties of determinants, from $(3.3)$ it follows that

[TABLE]

[TABLE]

hence the last determinant is $D!\cdot G_{D}(x_{1},\ldots,x_{D})$. Note that (3.8) implies that

[TABLE]

From (3.9) and (3.10) it follows that $F_{1,D}(x_{1},\ldots,x_{D})=$

[TABLE]

Also, from (3.8), we have $B_{\ell}(0,\ldots,0)=\binom{\ell}{j-1}B_{\ell-j+1}$, hence, from (3.5) and (3.11), we get

[TABLE]

Since $M_{D}$ is the determinant of a lower Hessenberg matrix, according to [4, pag.222,Theorem], we have the recursive relation

[TABLE]

We prove that

[TABLE]

using induction on $D\geq 1$. For $D=1$ we have $M_{1}=B_{1}=-\frac{1}{2}$, hence the (3.14) holds. If $D\geq 2$ then from induction hypothesis and (3.14) it follows that

[TABLE]

Since $\binom{D+1}{\ell}=\frac{D+1}{\ell}\binom{D}{\ell-1},(\forall)1\leq\ell\leq D$, from (3.15) it follows that

[TABLE]

On the other hand

[TABLE]

hence (3.16) completes the induction step. Therefore, we proved $(\ref{wish})$ and thus

[TABLE]

∎

For any integer $n\geq 1$, we denote

[TABLE]

the elementary symmetric polynomials in $\mathbb{Q}[x_{1},\ldots,x_{n}]$.

Theorem 3.4**.**

With the above notations, we have that

[TABLE]

Proof.

We use induction on $D\geq 1$. For $D=1$ we have

[TABLE]

hence the required formula holds. For $D\geq 2$, from (3.3) it follows that

[TABLE]

where $\widehat{x_{k}}$ means that the variable $x_{k}$ is omitted. From the induction hypothesis and (3.17) it follows that

[TABLE]

The relation (3.18) is equivalent to

[TABLE]

From (3.19), in order to complete the proof it is enough to show that

[TABLE]

Since $B_{D}(x_{k})=\sum_{s=0}^{D}\binom{D}{s}B_{D-s}x^{s}$, it follows that (3.20) is equivalent to

[TABLE]

for any $0\leq t\leq D$. Since, by Proposition $3.3(3)$, we have that

[TABLE]

it is enough to prove (3.21) for $0\leq t\leq D-1$. Similarly, by Proposition $3.2(2)$ we can dismiss the case $t=0$. Assume in the following that $1\leq t\leq D-1$. As the both sides in $(\ref{iedu})$ are symmetric polynomials, it is enough to prove that $(\ref{iedu})$ holds when we evaluate it in $x_{D-t+1}=\cdots=x_{D}=0$. Moreover, in this case, $E_{D-1,D-1-\ell}(x_{1},\ldots,\widehat{x_{k}},\ldots,x_{D})=0$ for any $\ell<t$. Therefore, $(\ref{iedu})$ is equivalent to

[TABLE]

hence it is equivalent to

[TABLE]

which can be easily proved by expanding a Vandermonde determinant of order $D-t$. ∎

Corollary 3.5**.**

We have that

[TABLE]

Proof.

From (3.2) and Theorem $3.4$ it follows that

[TABLE]

On the other hand

[TABLE]

hence, from (3.22) and (3.23) we get the required result. ∎

Unfortunately, in the general, it seems to be very difficult to give an exact formula for $F_{r,D}(x_{1},\ldots,x_{D})$. What it is easy to show is the following generalization of Proposition $3.3(1)$.

Proposition 3.6**.**

For any integers $r,D\geq 1$, there exists a symmetric polynomial $G_{r,D}$ of degree $\leq r^{2}\binom{D+1}{2}-r\binom{D}{2}$ such that

[TABLE]

where, with the notations from (3.7), we have that

[TABLE]

Proof.

Using standard properties of determinants, as in the proof of formula $(3.9)$, we get the required decomposition. The fact that $G_{r,D}(x_{1},\ldots,x_{D})$ is symmetric follows from the identity

[TABLE]

and the decomposition $F_{r,D}(x_{1},\ldots,x_{D})=\prod_{1\leq i<j\leq D}(x_{j}-x_{i})^{r}G_{r,D}(x_{1},\ldots,x_{D})$. ∎

4 An approach to compute $p_{\mathbf{a}}(n)$

Let $\mathbf{a}:=(a_{1},a_{2},\ldots,a_{r})$ be a sequence of positive integers, $r\geq 1$. Let $D$ be a common multiple of $a_{1},\ldots,a_{r}$. Using the notations and definitions from the second section, according to [7, Proposition $2.4$] and (2.3), the function $\zeta_{\mathbf{a}}(s)$ is meromorphic in the whole complex plane with poles at most in the set $\{1,\ldots,r\}$ which are all simple with residues

[TABLE]

On the other hand, according to [7, Theorem $2.10$] or [11, Formula (3.9)] and (2.2), we have that

[TABLE]

It follows that

[TABLE]

On the other hand, from $(1.9)$ it follows that

[TABLE]

If we see $d_{\mathbf{a},m}(v)$ as indeterminates, (4.3) and (4.4) form a system of linear equations with the determinant

[TABLE]

From (4.5) and the identity $B_{n}(1-x)=(-1)^{n}B_{n}(x)$ it follows that

[TABLE]

Remark 4.1**.**

Similarly to Proposition $2.1$, if $\bar{\Delta}_{r,D}\neq 0$, then $d_{\mathbf{a},m}(v)$, $0\leq m\leq r-1$, $1\leq v\leq D$ are the solutions of the system of linear equations consisting in $(4.3)$ and $(4.4)$.**

Now we consider the polynomial $\bar{F}_{r,D}\in\mathbb{Q}[x_{1},\ldots,x_{D}]$ defined as

[TABLE]

From (4.6) and (4.7) it follows that

[TABLE]

Note that if $D=1$ then (4.5) and (4.7) implies

[TABLE]

therefore, in the following we assume $D\geq 2$.

Using elementary operations in (4.7) and the notations (3.7) it follows that $\bar{F}_{r,D}(x_{1},\ldots,x_{D})=$

[TABLE]

[TABLE]

We denote the last determinant in (4.9) with $\bar{G}_{r,D}(x_{1},\ldots,x_{D})$ and we note that $\bar{G}_{r,D}$ is a symmetric polynomial with

[TABLE]

Proposition 4.2**.**

For any $D\geq 2$ we have that

[TABLE]

Proof.

(1) Using the method from [9, Page 262], we get

[TABLE]

[TABLE]

(2) The last identity follows $(1)$, $(4.8)$ and $(4.9)$. ∎

Remark 4.3**.**

For $D=2$, according to (4.9) we have that

[TABLE]

[TABLE]

On the other hand, according to (3.8), we have that

[TABLE]

In particular, from (4.12) and (4.13) it follows that

[TABLE]

From Lemma $3.1$ and (4.14) it follows that

[TABLE]

Our computer experiments in Singular [6] and Remark $4.3$ yield us to the following:

Conjecture 4.4**.**

For any $r\geq 1$, it holds that

[TABLE]

We checked Conjecture $4.4$ for $r\leq 4$ and we are convinced that the formula holds in general. Our computer experiments in Singular [6] yield us also to the following:

Conjecture 4.5**.**

For any $D\geq 2$, it holds that

[TABLE]

where $K(D)\in\mathbb{Q}$. Moreover, $K(D)\neq 0$, hence $\bar{\Delta}_{2,D}\neq 0$.

We checked Conjecture $4.5$ for $D\leq 4$ and we believe it is true in general. Unfortunately, we are not able to “guess” a general formula for $\bar{F}_{r,D}$, the situation being wild even for $D=r=3$ as $\bar{G}_{3,3}$ is an irreducible polynomial of degree $18$.