Riordan Matrix Representations of Euler’s Constant and Euler’s Number

Abstract

We show that the Euler-Mascheroni constant and Euler’s number can both be represented as a product of a Riordan matrix and certain row and column vectors.

Dedicated to David Harold Blackwell (April 24, 1919–July 8, 2010)

1. Introduction

It was shown by Kenter [1] that the Euler-Mascheroni constantcan be represented as a product of an infinite-dimensional row vector, the inverse of a lower triangular matrix, and an infinite-dimensional column vector:Kenter’s proof uses induction, definite integrals, convergence of power series, and Abel’s Theorem. In this paper, we recast this statement using the language of Riordan matrices. We exhibit another proof as well as a generalization. Our main result is the following.

Theorem 1. Consider sequences , , and of complex numbers such that , as well as an integer exponent . Assume that (i)the power series , , , and are convergent in the interval ;(ii)the following complex residue exists: Then, the matrix product is equal to the above residue.

The infinite-dimensional lower triangular matrix is an example of a Riordan matrix. Specifically, it is that Riordan matrix associated with the power series . Kenter’s result follows by careful analysis of the power series:The coefficients are sometimes called the “logarithmic numbers” or the “Gregory coefficients”; these are basically the Bernoulli numbers of the second kind up to a choice of sign. (Kenter employs the coefficients .) The idea of this paper is that we have the matrix productwhich is equivalent to the recursive identity , which is valid whenever . The matrix product, and hence the recursive identity, can be derived from properties of Riordan matrices. Kenter’s result follows from the identity , which in turn follows from an identity involving a definite integral.

As another consequence of our main result, we can also show that Euler’s numbercan be represented as a product of an infinite-dimensional row vector, a lower triangular matrix, and an infinite-dimensional column vector.

Corollary 2. For any integers , , and with , the number is equal to the matrix product

In the process of proving these generalizations, we present a representation theoretic view of Riordan matrices. That is, we consider the matrices as representations of a certain group , namely, the Riordan group, acting on an infinite-dimensional vector space , namely, the collection of those formal power series in , where .

2. Introduction to Riordan Matrices

We wish to list several key results in the theory of Riordan matrices. To do so, we recast this theory using techniques from representation theory very much in the spirit of Bacher [2]. Our ultimate goal in this section is to explain how Riordan matrices are connected to a permutation representation of a certain group acting on an infinite-dimensional vector space . Some of the notation in the sequel will differ from standard notation such as that given by Shapiro et al. [3] and Sprugnoli [4, 5], but we will explain the connection.

2.1. Group Actions

Before developing the representation theoretic view, we give the definition of a Riordan matrix and few related useful properties. Let be a field; it is customary to set as the set of complex numbers, but, in practice, is the set of rational numbers. Set as the collection of formal power series in an indeterminate ; we will view this as a -vector space with countable basis . For most of this article, we will not be concerned with regions of convergence for these series.

There are three binary operations which will be of importance to us, namely, multiplication , composition , and addition +. Explicitly, if we writethen we have the formal power seriesThere are three subsets of the vector space which will be of interest to us in the sequel.

Proposition 3. Define the subsets (i) is a group under multiplication , is a group under composition , and is a group under addition +. In particular, is a -vector space with countable basis .(ii)The map which sends to the automorphism is a group homomorphism, where is the compositional inverse of . In particular, is a group under the binary operation defined by (iii)The map defined by is a group action of on .

We use to denote the compositional inverse so that we will not confuse this with the multiplicative inverse . Later, we will show that is isomorphic to the Riordan group . Moreover, we will show that , a normal subgroup of , is isomorphic to the Appell subgroup of . The motivation of this result is to use the action of on to write down a permutation representation and then use the canonical basis of to list infinite-dimensional matrices.

Proof. We show (i) to fix some notation to be used in the sequel. Since for any , we see that is an associative binary operation. The identity is the constant power series , and the inverse of is its reciprocal, seen to be a power series by expressing said reciprocal in terms of a formal geometric series:Since and for any , we see that is an associative binary operation. The identity is the power series , and the inverse of is its compositional inverse having the implicitly defined coefficientsSince for any , we see that is an associative binary operation. The identity is the constant power series , and the inverse of is the negation , seen to be a power series with .
Now, we show (ii). Since for any and , we see that is well defined. Given , we have because for all we haveHence, is indeed a group homomorphism. The semidirect product consists of pairs with and , where the binary operation is defined byFinally, we show (iii). The map is defined as the formal identitySince , we see that the map is well defined. As the identity element of is , we see that so that it acts trivially on . Given two elements and , we have the identitySimilarly, given two elements and , we have the identityHence, is indeed a group action.

2.2. Riordan Matrices

Recall that the setis a -vpng . Since the semidirect product acts on , we have a “permutation” representation . Explicitly, this representation is defined on the basis elements of via the formal identity(Recall that is the compositional inverse of .) The matrix with respect to the basis is given by the lower triangular matrixRecall that yet . The following result explains the main multiplicative property of these matrices.

Theorem 4. Continue notation as above. (i) is a group homomorphism. That is, (ii)For a generating function with ,

Such matrices are called the Riordan matrices associated with the pair . The collection of Riordan matrices is a group which is isomorphic to ; this is the Riordan group. The collection of matrices is a group which is isomorphic to ; this normal subgroup is the Appell subgroup of .

Proof. We show (i). In the proof of Proposition 3, we found that for each we have the following formal identity involving power series as elements of :In particular, this holds for the basis elements , so the result follows.
Now, we show (ii). For a generating function , we have the productso matrices in the Appell subgroup are in the formThis gives the matrix productso the result follows.

2.3. Examples

Let . Using elementary calculus, we find the power series expansionswhich are valid whenever . Hence, the formal power seriesis an element of and has multiplicative inverseWe have the productwhich yields the matrixSimilarly, we have the productSince we may use Theorem 4 to conclude that , we find the identityThese matrices are elements of the Appell subgroup of .

2.4. Relation with Standard Notation

Standard references for Riordan matrices are Shapiro et al. [3] and Sprugnoli [4, 5]. The notation employed above is not the typical one, so we explain the connection. Consider sequences and of complex numbers , where . Upon associating generating functions and with these sequences, respectively, the standard notation for a Riordan matrix is that infinite-dimensional matrix given byin terms of the compositional inverse of . Indeed, the entry in the th row and th column satisfies the relationas formal power series in . Equivalently, a Riordan matrix can be defined by a pair of generating functions.

Corollary 5 (fundamental theorem of the Riordan group [3, 5, 6]). Continue notation as above. (i)The product of Riordan matrices is again a Riordan matrix. Explicitly, their product satisfies the relation (ii)For a generating function with , one has the product

Proof. Statement (i) is shown in [3, Eq. 5] and [6, Proof of Thm. 2.1], but we give an alternate proof. Upon denoting and for and , we find the matrix productwhich follows directly from Theorem 4. Statement (ii) is also shown in [6], but it follows directly from Theorem 4 as well.

3. Proof of Kenter’s Result and Generalizations

3.1. Main Result

We now prove Theorem 1.

Proof of Theorem 1. With the three power series , , and convergent in the interval , consider the power seriesAs elements of the Appell subgroup of , we invoke Theorem 4 to see that we have the matrix product . In particular, the first column is given byHence, the matrix productis equal to the sum . We wish to evaluate this sum using complex analysis.
By assumption, the power series , , and are convergent in the interval . Hence, for each fixed real number satisfying , the functions and are uniformly convergent inside a closed disk . Hence, we can interchange summation and integration to find the integral around the boundary to be equal toHere, is the complex conjugate of . As , the integral exists, so by Cauchy’s Residue Theorem it must be equal toThe theorem follows upon equating this with (44).

3.2. Applications

We explain how to use Theorem 1 in order to express Euler’s number in terms of Riordan matrices.

Proof of Corollary 2. The coefficients of the matrices in (9) correspond to the three power seriesFor a complex number with , we have the identityThe residue corresponds to the coefficient of the term, so we consider the terms where :The corollary follows now from Theorem 1.

Kenter’s result is also an application of Theorem 1.

Corollary 6 (see [1]). The Euler-Mascheroni constant is equal to the matrix product

Proof. The coefficients of the matrices above correspond to the three power seriesWe will choose the exponent . We will express the reciprocal as the power serieswhich is also convergent in the interval . (Recall that the coefficients are sometimes called the “logarithmic numbers” or the “Gregory coefficients.”) For a complex number with , we have the identityThe residue corresponds to the coefficient of the term, so we consider the terms where :The corollary follows now from Theorem 1.

We conclude by stating that Theorem 1 can also be used to show Riordan matrix representations for and . Finding matrix representations of other constants, like , , and the Golden Ratio , is of interest.

Disclosure

Both authors gave the recent annual Blackwell Lectures, organized by the National Association of Mathematicians (NAM) as part of the MAA MathFest. The first author gave his presentation during the summer of 2009, whereas the second gave his during the summer of 2010.

Competing Interests

The authors declare that they have no competing interests.