The purpose of this note is to provide an expository introduction to some more curious integral formulas and transformations involving generating functions. We seek to generalize these results and integral representations which effectively provide a mechanism for converting between a sequence’s ordinary and exponential generating function (OGF and EGF, respectively) and vice versa. The Laplace transform provides an integral formula for the EGF-to-OGF transformation, where the reverse OGF-to-EGF operation requires more careful integration techniques. We prove two variants of the OGF-to-EGF transformation integrals from the Hankel loop contour for the reciprocal gamma function and from Fourier series expansions of integral representations for the Hadamard product of two generating functions, respectively. We also suggest several generalizations of these integral formulas and provide new examples along the way.

Given a sequence

Notice that we can always construct these functions over any sequence

We can precisely define the form of an integral transformation (in one variable) as [

Generating function transformations form a useful combinatorial and analytic method (depending on perspective) which can be combined and employed to study new sequences of many forms. Our focus in this article is to motivate the constructions of generating function transformations as meaningful and indispensable tools in enumerative combinatorics, combinatorial number theory, and in the theory of partitions, among other fields where such applications live. The particular modus operandi within this article shows the evolution of integral transforms for the reciprocal gamma function, and its multi-factorial integer sequence special cases, as a motivating method for enumerating several types of special sequences and series which we will consider in the next sections.

The references [

A time consuming hobby that the author assumes from time to time is rediscovering old and unusual identities in mathematics textbooks– particularly in the areas of combinatorics and discrete mathematics. Favorite books to search include Comtet’s

In this direction, we have an easy conversion integral for converting from the EGF of a sequence

That being said, Graham, Knuth and Patshnik already suggest a curious “known” integral formula for performing this corresponding OGF-to-EGF conversion operation of the following form [

The statement of this result is given without proof in the identity-full appendix section of the textbook. When first (re)-discovered many years back, the author assumed that the motivation for this integral transformation must correspond to the non-zero paths of a complex contour integral for the reciprocal gamma function. For many years the precise formulation of a proof of this termwise integral formula and its generalization to enumerating terms of reciprocal generalized multi-factorial functions, such as

Integral transformations are a powerful and convenient formal and analytic tool which are used to study sequences and their properties. Moreover, they are easy to parse and apply in many contexts with only basic knowledge of infinitesimal calculus making them easy-to-understand operations which we can apply to sequence generating functions. The author is an enthusiast for particularly pretty or interesting integral representations (cf. [

One notable example of such an integral transformation given in [

Another source of generating function transformation identities correspond to the bilateral series given by Lindelöf in [

Additional series transformations involving a sequence generating function into the form of

Applications of these square series integral representations include many new integral formulas for theta functions and classical

There are more general Meinardus methods for computing asymptotics of the coefficients of classes of partition number generating functions of the form [

In this short note we provide proofs of known integral formulas providing an ordinary-to-exponential generating function operation. We prove the following theorem using the Hankel loop contour for the reciprocal gamma function in

We also give a rigorous proof of the next integral formula relating

The proof of Theorem 2 is given in

Since

For

The coefficients

We seek an exact integral representation for the reciprocal gamma function, not just an integral formula defining the coefficients of its Taylor series expansion about zero in this case. To find such a formula we must use the Hankel loop contour

Working from the figure, we have that [

We will first approach the contribution of the section of the contour given by

When we take the first small-order limits we obtain

We then finally arrive at the stated known integral formula for the reciprocal gamma function which holds for any fixed real

Since we are initially motivated by finding a general conversion integral from a sequence OGF into its EGF, we notice that we require an application of (

We can perform the same “trick” of the generating function trades to sum a “doubly exponential” sequence generating function when we replace the sequence OGF by its EGF in the previous equation:

Perhaps at first glance this iterated integral formula is somewhat unsatisfying since we have really just repeated the procedure for constructing the first integral twice, but in fact there are notable special case applications which we can derive from this method of summation which provide new integral representations for otherwise hard-to-sum hypergeometric series.

For example, if we take the geometric series sequence case where

There is an integral representation for this function which is simpler to evaluate in the general case given in (

One curious identity that the author has come across relating the OGF of a sequence to its EGF is found in the appendices of the Concrete Mathematics reference [

Finding a precise method of verifying this unproven identity is the initial motivation for this note. Given the discussion and lead up to an integral for the reciprocal gamma function taken over the real line via the Hankel loop contour in the last section, the author initially assumed—and asked with no replies in online math forums—that this computationally correct integral representation must correspond to the non-zero components of some complex contour integral. It turns out that this formula follows from the basic theory and constructions of Fourier analysis.

Given a sequence,

The change of variables

Another satisfyingly less analytical and more formally motivated explanation for this behavior can be given by considering known integral formulas for the Hadamard product of two series given in terms of the orthogonal set

The primary goal of the first post [

In the spirit of our realization that the integral representation in (

The modified exponential series of the first type identified above are primarily summed in closed-form using expansions of the Mittag–Leffler functions,

We have proved two key new forms of integral representations for the reciprocal gamma function on the real line. By composition and the uniform convergence of power series for functions defined on some disc

We have provided several examples of motivating cases of our so-termed generating function transformations by integral-based methods in

This research received no external funding.

The author declares no conflict of interest.

The Hankel loop contour providing an integral representation of the reciprocal gamma function when