# An explicit form of the polynomial part of a restricted partition function

- Karl Dilcher
^{1}and - Christophe Vignat
^{2, 3}Email author

**Received: **31 August 2016

**Accepted: **17 November 2016

**Published: **5 January 2017

## Abstract

We prove an explicit formula for the polynomial part of a restricted partition function, also known as the first Sylvester wave. This is achieved by way of some identities for higher-order Bernoulli polynomials, one of which is analogous to Raabe’s well-known multiplication formula for the ordinary Bernoulli polynomials. As a consequence of our main result we obtain an asymptotic expression of the first Sylvester wave as the coefficients of the restricted partition grow arbitrarily large.

## Keywords

## Mathematical Subject Classification

## 1 Background

*restricted partitions*, where the following question has been studied quite extensively: given a vector \(\mathbf{d }:=(d_1,d_2,\ldots ,d_m)\) of positive integers, let \(W(s,\mathbf{d })\) be the number of partitions of the integer

*s*with parts in \(\mathbf{d }\), i.e., \(W(s,\mathbf{d })\) is the number of solutions of

*j*of the components of \({\mathbf{d }}\). Sylvester [23] showed that for each such

*j*, \(W_j(s,{\mathbf{d }})\) is the coefficient of \(t^{-1}\), i.e., the residue, of the function

*j*th root of unity, for instance \(\rho _j=e^{2\pi i/j}\), and where we set \(\gcd (0,0)=1\) by convention. In other words, the sum in (4) is taken over all primitive

*j*th roots of unity \(\rho _j^{\nu }\). These

*Sylvester waves*have been studied in great detail in recent years; see, e.g., [10, 19, 20]; see also [7, 8, 14] for a broader perspective, and [21] for computations related to restricted partitions.

For \(j=1\), the right-hand side of (4) is recognizable as being very close to the generating function of a higher-order Bernoulli polynomial. This fact was used by Rubinstein and Fel [20] to write \(W_1(s,{\mathbf{d }})\) in a very compact form in terms of a single higher-order Bernoulli polynomial [see (33) below]. A version of this result, given in two different forms, was earlier obtained by Beck, Gessel and Komatsu [3], as mentioned in [20]. Similarly, for \(j=2\) we have \(\rho _j=-1\), and the right-hand side of (4) will typically lead to a convolution sum of higher-order Bernoulli and higher-order Euler polynomials; this was also done in [20]. Furthermore, Rubinstein and Fel extended this approach and expressed \(W_j(s,{\mathbf{d }})\) for arbitrary *j* in terms of generalized Eulerian polynomials of higher order, in addition to the expected higher-order Bernoulli polynomials.

In a subsequent paper, Rubinstein [19] showed that all the Sylvester waves \(W_j(s,{\mathbf{d }})\) can be written as linear combinations of the first wave (\(j=1\)) alone, with modified integers *s* and vectors \({\mathbf{d }}\) [see (47) below]. This makes it worthwhile to give further consideration to \(W_1(s,{\mathbf{d }})\), which is the purpose of the present paper. Our main result is the following explicit formula for \(W_1(s,{\mathbf{d }})\); its significance lies in the fact that it does not contain Bernoulli numbers or polynomials.

### Theorem 1.1

For a more compact form of this identity, see Sect. 5.

Towards proving this theorem, we derive (or re-derive) some identities which are analogous to classical results in the theory of Bernoulli polynomials and their higher-order analogues. Our main tool is a symbolic notation which, in spite of some similarities, is different from the classical umbral calculus. This will be introduced in Sect. 2, and we apply it in Sect. 3 to prove the auxiliary results as well as Theorem 1.1. In Sect. 4 we present some examples and consequences of Theorem 1.1, including an asymptotic expression. We finish this paper with some additional remarks in Sect. 5.

## 2 Symbolic notation

Although the results in Sect. 3 could also be proved (and in some cases have been proved) by other methods, especially using generating functions, the symbolic notation described below makes the discovery and proof of some identities considerably easier. While there are similarities to the classical umbral calculus (see, e.g., [11] or [18]), our notation is more specific to Bernoulli numbers and polynomials, and is related to probability theory. The following brief exposition is partly taken from [6]; we repeat it here for the sake of completeness.

*Bernoulli symbol*\({\mathcal {B}}\) by

*n*th Bernoulli number. So, for instance, we can be rewrite the usual definition for the Bernoulli polynomial \(B_n(x)\),

*uniform symbol*\({\mathcal {U}}\) is defined by

*f*is an arbitrary polynomial. From (11) we immediately obtain, in analogy to (6),

*f*.

*discrete uniform symbol*\({\mathcal {U}}_{\{a_1,\ldots ,a_m\}}\) by way of the generating function

*f*, which can be seen as a discrete analogue of (11). From the definition (15) we immediately obtain the identity

*higher-order Bernoulli symbol*\({\mathcal {B}}^{(k)}\) by

## 3 Higher-order Bernoulli polynomials and proof of Theorem 1.1

*Bernoulli polynomial of order*

*k*is defined by the generating function

### Theorem 3.1

*n*,

*m*, and \(d_1,\ldots ,d_m\) be positive integers, and set \(d:=d_1\ldots d_m\). Then

### Proof

*m*by \(\widetilde{d}_i\), we get

*m*, (26) leads to (24), and we are done. \(\square \)

We note that this result is not new. In fact, it is a special case of identity (60) in the classical book of Nörlund [13, p. 135]. However, the method of proof in [13] is very different from ours and relies on the theory of finite differences. On the other hand, it should also be mentioned that the symbolic notation involving the Bernoulli symbol, more or less as used on the left-hand side of (24), can also be found in [13], on p. 135 and elsewhere.

*Bernoulli-Barnes polynomials*.

From (24) and (28) we can now obtain the following analogue of (28).

### Corollary 3.2

Before proving this, we note that in the case \(d_1=\cdots =d_m=1\), the multiple sum on the right of (31) collapses to the single term \(\ell _1=\dots =\ell _m=0\), and (31) reduces to (28).

### Proof of Corollary 3.2

*d*, we obtain (31). \(\square \)

For the proof of Theorem 1.1 we also need the following reflection formula, which can be found in [13, p. 134]. For the sake of completeness we will provide a proof.

### Lemma 3.3

*m*and \(d_1,\ldots ,d_m\) be positive integers, and let \({\mathbf{d }}:=(d_1,\ldots ,d_m)\) and \(\sigma :=d_1+\cdots +d_m\). Then for all \(n\ge 0\) we have

### Proof

## 4 Examples and consequences of Theorem 1.1

Theorem 1.1 can also be used to determine the two highest coefficients of \(W_1(s,{\mathbf{d }})\); we state this as a corollary.

### Corollary 4.1

The leading coefficient, which has long been known (see, e.g., [3] and references therein), follows immediately from (5) if we note that by (27) the number of summands in the *m*-fold sum is \(d^{m-1}\). The second coefficient follows from a simple expansion of the product on the right of (5). We skip the details since both coefficients follow immediately from the identity (3) in [3].

For the special case \({\mathbf{d }}=(1,2,\ldots ,m)\), the identity (40) can be found in [17, Satz 1] and [24, p. 311].

From the fact that the first Sylvester wave is a polynomial, it is clear that for fixed *m* and bounded components of \({\mathbf{d }}\), \(W_1(s,{\mathbf{d }})\) is asymptotically equal to the leading term in (40), as *s* gets arbitrarily large. However, it is not immediately clear what happens if the components of \({\mathbf{d }}\) also grow, along with *s*. This is addressed in the following consequence of Theorem 1.1.

### Corollary 4.2

*d*grow arbitrarily large in such a way that at least two of the components \(d_j\), \(1\le j\le m\), are unbounded. Then

*d*.

The proof of this result relies on interpreting the *m*-fold sum on the right of (5) as a Riemann sum of a certain multiple integral. We therefore begin by evaluating this integral.

### Lemma 4.3

### Proof

*m*-fold) use of the cancellation property (14). \(\square \)

### Proof of Corollary 4.2

*d*grows, then \(\Delta x_1=1/(d_2\ldots d_m)\) would not approach 0 as

*d*grows. However, this cannot happen if at least two of the \(d_j\) are unbounded as

*d*grows. If we now multiply the sum on the right of (43) by

*m*-fold sum as a Riemann sum that converges to the integral in (42). Hence we have, by Lemma 4.3,

*s*/

*d*. \(\square \)

## 5 Additional Remarks

1. If we divide each factor in the product on the right of (5) by *d*, we see that the resulting product can be written as a Pochhammer symbol (rising factorial) or as a falling factorial. But we can also combine it with \((m-1)!\) in the denominator; using the (generalized) binomial coefficient \(\left( {\begin{array}{c}x\\ n\end{array}}\right) =x(x-1)\ldots (x-n+1)/n!\) we can then rewrite Theorem 1.1 as follows.

### Corollary 5.1

The binomial coefficient on the right of (44) is reminiscent of some combinatorial objects related to partitions and compositions. (Note, however, that \((s-\ell )/d\) is generally not an integer).

While this is not the same as the number of compositions of *s* into *m* parts, there is a connection: The number of compositions of *n* into exactly *m* parts, each at least *k*, is \(\left( {\begin{array}{c}m-1+n-km\\ m-1\end{array}}\right) \); see [1, p. 63].

2. For the sake of completeness we cite the main result of Rubinstein [19], which we referred to in the Introduction. While for \(W_1(s,{\mathbf{d }})\) the order of the components in the given vector \({\mathbf{s }}=(d_1,\ldots ,d_m)\) is irrelevant, this becomes an issue for the Sylvester waves \(W_j(s,{\mathbf{d }})\) when \(j\ge 2\).

*j*divides the first \(k_j\) components. We denote

*j*.

*prime radical circulator*\(\psi _j(s)\), which for positive integers

*s*is defined by

*j*th root of unity. For prime

*j*we have

*j*; see [19] and the references therein for further details.

### Example 1

If *j* does not divide any of the components of \({\mathbf{d }}\), then \(k_j=m\), and \(W_j(s,{\mathbf{d }}) = 0\) as the sum on the right of (47) is empty. This is consistent with (3) and the statement following it.

### Example 2

## Declarations

### Author's contributions

Both authors read and approved the final manuscript.

### Open Access

This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

### Acknowledgements

The first author was supported in part by the Natural Sciences and Engineering Research Council of Canada.

Dedicated to the memory of our friend and colleague Jonathan M. Borwein, mathematician extraordinaire.

### Competing interests

The authors declare that they have no competing interests.

## Authors’ Affiliations

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