= 10000

Keywords: integer partitions, Sylvester waves.

2010 Mathematics Subject Classification: 11P82. The problem of partitions of positive integers has long history started from the work of Euler who laid a foundation of the theory of partitions [1], introducing the idea of generating functions. A. Cayley [2] and J.J. Sylvester [3] provided a new insight and made a remarkable progress in this field. Sylvester found [3], [4] the procedure for computation of a restricted partition function, while Cayley described the symmetry properties of these functions. The restricted partition function \(W(s,{\boldsymbol{d}}^m) \equiv W(s,\{d_1,d_2,\ldots,d_m\})\) is a number of partitions of \(s\) into positive integers \(\{d_1,d_2,\ldots,d_m\}\). Sylvester showed that the partition function can be split into polynomial and quasiperiodic parts and presented as a sum of components called the Sylvester waves \[W(s,{\boldsymbol{d}}^m) = \sum_{j=1} W_j(s,{\boldsymbol{d}}^m)\;, \label{SylvWavesExpand}\tag{1}\] where summation runs over all divisors of the elements in the set \({\boldsymbol{d}}^m\). The wave \(W_j(s,{\boldsymbol{d}}^m)\) is a quasipolynomial in \(s\) closely related to prime roots \(\rho_j\) of unity. Namely, Sylvester showed [4] that the wave \(W_j(s,{\boldsymbol{d}}^m)\) is a coefficient of \({t}^{-1}\) in the series expansion in ascending powers of \(t\) of \[F_j(s,t)=\sum_{\rho_j} \frac{\rho_j^{-s} e^{st}}{\prod_{k=1}^{m} \left(1-\rho_j^{d_k} e^{-d_k t}\right)}\;, \quad \rho_j=\exp(2\pi i n/j), \label{generatorWj}\tag{2}\] where the summation is made over all prime roots of unity \(\rho_j\) for \(n\) relatively prime to \(j\) (including unity) and smaller than \(j\). The relation (2 ) is just a recipe for calculation of the partition function and it does not provide an explicit expression for \(W_j(s,{\boldsymbol{d}}^m)\).

Using the Sylvester recipe we found in [5] a formula for the Sylvester wave \(W_j(s,{\boldsymbol{d}}^m)\) as a finite sum of the Bernoulli polynomials of higher order [6], [7] multiplied by a periodic function of integer period \(j\). The polynomial part \(W_1(s,{\boldsymbol{d}}^m)\) of the partition function reads \[W_1(s,{\boldsymbol{d}}^m) = \frac{sec:1}{(m-1)!\;\pi_m} B_{m-1}^{(m)}(s + s_m, {\boldsymbol{d}}^m)\;, \quad s_m = \sum_{i=1}^m d_i, \;\;\pi_m = \prod_{i=1}^m d_i\;, \label{W951}\tag{3}\] with the Bernoulli polynomials of higher order are defined by the generating function [7]: \[\frac{e^{st} t^m \pi_m}{\prod_{i=1}^m (e^{d_it}-1)} = \sum_{n=0}^{\infty} B^{(m)}_n(s,{\boldsymbol{d}}^m) \frac{t^{n}}{n!}\;, \label{genfuncBernoulli0}\tag{4}\] and some of their properties are described in the Appendix. The Sylvester wave \(W_j(s,{\boldsymbol{d}}^m)\) for \(j>1\) can be written as \[\begin{align} W_j(s,{\boldsymbol{d}}^m) &=& \frac{j^{k_j-m}}{(m-1)! \; \pi_{m}} \sum_{n=0}^{k_j-1} C^{m-1}_n B_n(s+s_m,{\boldsymbol{d}}^{k_j}) \nonumber \\ &\times& \sum_{{\boldsymbol{r}}=0}^{j-1} B^{(m)}_{m-n-1}({\boldsymbol{r}}\cdot{\boldsymbol{d}}^{m-k_j},j{\boldsymbol{d}}^{m-k_j}) \Psi_j(s+s_m+{\boldsymbol{r}}\cdot{\boldsymbol{d}}^{m-k_j}), \label{WjOrig} \end{align}\tag{5}\] where we introduce two subsets – the subset \({\boldsymbol{d}}^{k_j}=\{d_{sec:1},d_{2},\ldots,d_{k_j}\}\) of the generators \(d_i\) divisible by \(j\) and the subset \({\boldsymbol{d}}^{m-k_j}=\{d_{k_j+1},d_{k_j+2},\ldots,d_{m}\}\) of those nondivisible by \(j\). The \((m-k_j)\)-dimensional vector \({\boldsymbol{r}}\) has the form \({\boldsymbol{r}} = \{r_{k_j+1},r_{k_j+2},\ldots,r_{m}\}\) with \(0 \le r_i \le j-1\). The prime circulator \(\Psi_j\) in (5 ) is the \(j\)-periodic function introduced in [2] that can be written as a sum of the simple periodic functions \[\Psi_j(s) = \sum_{\rho_j} \rho_j^s = \sum_{n} \Psi_{j,n}(s), \quad \Psi_{j,n}(s) = \exp(2\pi i n s/j), \quad \sum_{k=0}^{j-1} \Psi_{j,n}(k) = 0, \label{gencirc}\tag{6}\] for all \(1 \le n \le j-1\) relatively prime to \(j\).

Considering (5 ) it was suggested in [5] to extend the outer summation to \(m-1\) that with the help of (17 ) enables to present \(W_j(s,{\boldsymbol{d}}^m)\) as follows \[\begin{align} W_j(s,{\boldsymbol{d}}^m) = \frac{j^{k_j-m}}{(m-1)! \; \pi_{m}} \sum_{{\boldsymbol{r}}=0}^{j-1} B^{(m)}_{m-1}(s+s_m+ {\boldsymbol{r}}\cdot{\boldsymbol{d}}^{m-k_j},{\boldsymbol{d}}^{m}_j) \Psi_j(s+s_m+{\boldsymbol{r}}\cdot{\boldsymbol{d}}^{m-k_j}), \label{WjFin} \end{align}\tag{7}\] where we introduce the modified set \({\boldsymbol{d}}^{m}_j = {\boldsymbol{d}}^{k_j} \cup j{\boldsymbol{d}}^{m-k_j}\). Comparing (7 ) to (3 ) we observe that the Sylvester wave \(W_j(s,{\boldsymbol{d}}^m)\) can be viewed as a weighted sum of the polynomial part with shifted argument with the weights given by the prime circulator \(\Psi_j\). This result is valid in the assumption that the added terms corresponding to the values of \(k_j \le n \le m-1\) in (5 ) sum up to zero. It was positively tested on multiple examples but the proof was not given. The aim of this manuscript is to present the formal proof of added terms vanishing which indicates that the form (7 ) is indeed correct.

Given positive integers \({\mu},{\nu},j\) define the integer vectors \({\boldsymbol{d}}=\{d_1,d_2,\ldots,d_{\mu}\}\), \({\boldsymbol{r}}=\{r_1,r_2,\ldots,r_{\mu}\}\) with positive \(d_i > 0\) and \(0 \le r_i \le j-1\). Introduce an arbitrary vector \({\boldsymbol{e}}=\{e_1,e_2,\ldots,e_q\}\) and the \(j\)-periodic function \(\Psi_{sec:j}(s)\) to define a quantity \[\sigma_{\nu}(s,t,{\boldsymbol{d}},{\boldsymbol{e}}) = \sum_{{\boldsymbol{r}}=0}^{j-1} B^{(q)}_{\nu}(t+{\boldsymbol{r}}\cdot{\boldsymbol{d}},{\boldsymbol{e}}) \Psi_{sec:j}(s+{\boldsymbol{r}}\cdot{\boldsymbol{d}}), \quad \sum_{k=0}^{j-1}\Psi_{sec:j}(k) = 0, \label{395a}\tag{8}\] where \(B^{(q)}_{\nu}(s,{\boldsymbol{e}})\) is the Bernoulli polynomial of the higher order satisfying the relations (16 ,17 ). Use (6 ) to rewrite \(\Psi_j(s+{\boldsymbol{r}}\cdot{\boldsymbol{d}})\) as the sum \[\Psi_j(s+{\boldsymbol{r}}\cdot{\boldsymbol{d}}) =\sum_{n} \Psi_{j,n}(s+{\boldsymbol{r}}\cdot{\boldsymbol{d}}) = \sum_{n}\Psi_{j,n}(s) \prod_{i=1}^{m} \Psi_{j,n}(r_i d_i), \quad\quad \sum_{k=0}^{j-1}\Psi_{j,n}(k) = 0. \label{395aa}\tag{9}\] The polynomial \(B^{(q)}_{\nu}(t+{\boldsymbol{r}}\cdot{\boldsymbol{d}},{\boldsymbol{e}})\) satisfies the relations \[B_{\nu}^{(q)}(t+{\boldsymbol{r}}\cdot{\boldsymbol{d}},{\boldsymbol{e}}) =\sum_{p=0}^{\nu} C_p^{\nu} B_{\nu-p}^{(q)}(t,{\boldsymbol{e}}) B_p({\boldsymbol{r}}\cdot{\boldsymbol{d}}), \quad B_{p}({\boldsymbol{r}}\cdot{\boldsymbol{d}}) =\sum_{k=0}^{p} C_k^{p} ({\boldsymbol{r}}\cdot{\boldsymbol{d}})^{k}B_{p-k}\;, \label{395b}\tag{10}\] where the factor \(({\boldsymbol{r}}\cdot{\boldsymbol{d}})^{k}\) can be written as \[({\boldsymbol{r}}\cdot{\boldsymbol{d}})^{k} = \left( \sum_{i=1}^{\mu} r_i d_i \right)^{k} = \sum_{\boldsymbol{l}} C^{k}_{\boldsymbol{l}}\prod_{i=1}^{\mu} (r_i d_i)^{l_i}, \quad C^{k}_{\boldsymbol{l}} = \frac{k!}{l_1!l_2!\ldots l_{\mu}!}, \quad \sum_{i=1}^{\mu} l_i = k, \label{395c}\tag{11}\] and \(C^{k}_{\boldsymbol{l}}\) denotes the multinomial coefficient. Substitute (11 ) into (10 ) to obtain \[B_{n}^{(q)}(t+{\boldsymbol{r}}\cdot{\boldsymbol{d}},{\boldsymbol{e}}) =\sum_{p=0}^{n} \sum_{k=0}^{p} C_p^{\nu}C_k^{p} B_{n-p}^{(q)}(t,{\boldsymbol{e}})B_{p-k} \sum_{\boldsymbol{l}} C^{k}_{\boldsymbol{l}}\prod_{i=1}^{\mu} (r_i d_i)^{l_i}\;. \label{395d}\tag{12}\] Use it together with (9 ) in the definition (8 ) \[\sigma_{\nu}(s,t,{\boldsymbol{d}},{\boldsymbol{e}}) = \sum_{p=0}^{\nu} \sum_{k=0}^{p} C_p^{\nu}C_k^{p} B_{\nu-p}^{(q)}(t,{\boldsymbol{e}})B_{p-k} \sum_{n}\Psi_{j,n}(s) \sum_{\boldsymbol{l}} C^{k}_{\boldsymbol{l}} \sum_{{\boldsymbol{r}}=0}^{j-1}\prod_{i=1}^{\mu} (r_i d_i)^{l_i} \Psi_{j,n}(r_i d_i). \label{395e}\tag{13}\] Note that for \(k < \mu\) all terms in the sum over \({\boldsymbol{l}}\) have at least one \(l_i\) equal to zero and the corresponding factor \((r_i d_i)^{l_i}\) is absent. In this case the sum over \({\boldsymbol{r}}\) \[\sum_{{\boldsymbol{r}}=0}^{j-1} \prod_{i=1}^{\mu} (r_i d_i)^{l_i} \Psi_{j,n}(r_i d_i) = 0,\] due to (9 ) and the corresponding contribution to \(\sigma_{\nu}(s,t,{\boldsymbol{d}},{\boldsymbol{e}})\) vanishes. It follows from (10 ) that the maximal value of \(k\) is equal to \(\nu\) and thus the function \(\sigma_{\nu}(s,t,{\boldsymbol{d}},{\boldsymbol{e}})\) satisfies \[\sigma_{\nu}(s,t,{\boldsymbol{d}},{\boldsymbol{e}}) = 0, \qquad 0 \le \nu \le \mu-1. \label{395f}\tag{14}\] Consider the inner sum in (5 ) that can be written as \[\sum_{{\boldsymbol{r}}=0}^{j-1} B^{(m)}_{m-n-1}({\boldsymbol{r}}\cdot{\boldsymbol{d}}^{m-k_j},j{\boldsymbol{d}}^{m-k_j}) \Psi_j(s+s_m+{\boldsymbol{r}}\cdot{\boldsymbol{d}}^{m-k_j}) = \sigma_{m-n-1}(s+s_m,0,{\boldsymbol{d}}^{m-k_j},j{\boldsymbol{d}}^{m-k_j}),\] so that \(\nu = m-n-1\) and \(\mu = m-k_j\). Use (14 ) to observe that for \[0 \le m-n-1 \le m-k_j-1 \quad \Rightarrow \quad k_j \le n \le m-1,\] the sum \(\sigma_{m-n-1}(s+s_m,0,{\boldsymbol{d}}^{m-k_j},j{\boldsymbol{d}}^{m-k_j})\) vanishes. This means that the addition to (5 ) the terms required to produce (7 ) does not change the value of the Sylvester wave \(W_j(s,{\boldsymbol{d}}^m)\).

Replacing in the r.h.s. of (7 ) all the factors \(\Psi_j(s+s_m+{\boldsymbol{r}}\cdot{\boldsymbol{d}}^{m-k_j})\) by unity and using the multiplication theorem (18 ) we find \[\begin{align} \frac{j^{k_j-m}}{(m-1)! \; \pi_{m}} \sum_{{\boldsymbol{r}}=0}^{j-1} B^{(m)}_{m-1}(s+s_m+ {\boldsymbol{r}}\cdot{\boldsymbol{d}}^{m-k_j},{\boldsymbol{d}}^{m}_j) = \frac{sec:1}{(m-1)! \; \pi_{m}} B^{(m)}_{m-1}(s+s_m,{\boldsymbol{d}}^{m}) = W_1(s,{\boldsymbol{d}}^m). \label{unit95weight} \end{align}\tag{15}\] This relation means that the Sylvester wave \(W_j(s,{\boldsymbol{d}}^m)\) represents the splitting of the polynomial part \(W_1(s,{\boldsymbol{d}}^m)\) of the partition function into a multiple sum of \(j\)-periodic contributions.

Appendix ↩︎

The symbolic technique for manipulating sums with binomial coefficients by expanding polynomials and then replacing powers by subscripts was developed in nineteenth century by Blissard, it is known as the umbral calculus [8]. An example of this notation is also found in [6] in section devoted to the Bernoulli polynomials \(B_k(x)\).

The well-known formulas for the ordinary Bernoulli polynomials can be written symbolically \[B_n(x) = \sum_{k=0}^{n} C^n_k B_k x^{n-k} \equiv (B+x)^n,\;\; B_n(x+y) = \sum_{k=0}^{n} C^n_k B_k(x) y^{n-k} \equiv (B(x)+y)^n, \label{A1}\tag{16}\] where after the expansion the exponents of \(B\) and \(B(x)\) are converted into the orders of the Bernoulli number and Bernoulli polynomial, respectively: \(B^k \Rightarrow B_k, B^k(x) \Rightarrow B_k(x)\).

Nörlund [7] introduced the Bernoulli polynomials of higher order having an umbral representation \[B_{n}^{(m)}(x,{\boldsymbol{d}}^m) = \left( d_m B + B^{(m-1)}(x,{\boldsymbol{d}}^{m-1}) \right)^n,\] that recursively reduces to a more symmetric form \[B_{n}^{(m)}(x,{\boldsymbol{d}}^m) = \left( x + d_1 B + d_2 B + \ldots + d_m B \right)^n = \left( x + \sum_{i=1}^m d_i B \right)^n. \label{A2}\tag{17}\] These polynomials satisfy [7] the multiplication theorem \[\sum_{r_1=0}^{m_1-1} \ldots \sum_{r_p=0}^{m_p-1} B^{(n)}_{k}(s+\sum_{i=1}^p \frac{r_id_i}{m_i},\{d_1,\ldots,d_n\}) = \prod_{i=1}^p m_i \cdot B^{(n)}_{k}(s,\{\frac{d_1}{m_1},\ldots,\frac{d_p}{m_p},d_{p+1},\ldots,d_n\}). \label{A3}\tag{18}\]

References↩︎