Inverse Factorial Expansions

• Inverse factorial expansions are series representations that express functions in terms of rising or falling factorials, capturing asymptotic behaviors not easily handled by power series.
• They rely on combinatorial structures such as Stirling and Bernoulli numbers to convert between power and factorial bases, providing analytic and numerical advantages.
• These expansions facilitate the summation of divergent series and offer explicit error bounds and convergence criteria for applications in special function analysis and mathematical physics.

An inverse factorial expansion represents a function as a series whose basis elements are descending (falling) or ascending (rising) factorials, often expressed as reciprocals of shifted products such as $z(z+1)\cdots(z+n)$. This structure is in contrast to standard power or Laurent series, and is particularly suited to capturing the asymptotics of functions with factorial-type growth or facilitating the summation of divergent power series. The coefficients of these expansions frequently involve combinatorial quantities such as Stirling numbers, Bernoulli numbers, and related families, providing both an analytic and combinatorial perspective on function expansions.

1. Classical Form and Fundamental Properties

A prototypical inverse factorial expansion for a function $f(z)$ is

$$f(z) = \sum_{n=0}^\infty \frac{a_n}{z(z+1)\cdots(z+n)} = \sum_{n=0}^\infty \frac{a_n}{(z)_{n+1}},$$

where $(z)_k = z(z+1)\cdots(z+k-1)$ is the Pochhammer symbol or rising factorial. Such expansions converge under mild growth conditions, determined by the behavior of $f(z)$ as $z\to\infty$, and the coefficients $\{a_n\}$ are uniquely specified under these conditions (Boyadzhiev, 2020).

The underlying connection to combinatorics arises via the Stirling numbers of the first kind, $s(n,k)$, which appear naturally in change-of-basis relations between powers and falling or rising factorials: $$x^{\underline{n}} = \sum_{k=0}^n s(n,k)\, x^k, \qquad x^m = \sum_{n=0}^m S(m,n)\,(x)_{n},$$ where $x^{\underline{n}} = x(x-1)\cdots(x-n+1)$ and $f(z)$0 are Stirling numbers of the second kind (Boyadzhiev, 2020, Schreiber, 2023).

The master identity enabling transition from Laurent to inverse-factorial bases is

$f(z)$1

Term-by-term application to a function's Laurent expansion at infinity yields explicit formulas for inverse factorial coefficients via the so-called Stirling transform: $f(z)$2 where $f(z)$3 are the coefficients in the ordinary Laurent expansion (Boyadzhiev, 2020).

2. Explicit Expansions for Special Functions

Inverse factorial series provide concrete, computationally useful representations for a wide class of transcendental and special functions, including but not limited to:

• Gamma Function: Expansions for $f(z)$4 and products of Gamma functions can be written in terms of falling factorials, derivative/Bell polynomials, and combinatorial coefficients involving Stirling numbers. For example (Rzadkowski, 2010, Karp et al., 2017):

$f(z)$5

where $f(z)$6 is a Stirling-number sum.

• General gamma ratios: For $f(z)$7, there exists an expansion (Karp et al., 2017):

$f(z)$8

with explicit recurrence/formulas for $f(z)$9 using Nørlund-Bernoulli polynomials.

• Zeta and Polylogarithm Functions: Formulas exist for $$f(z) = \sum_{n=0}^\infty \frac{a_n}{z(z+1)\cdots(z+n)} = \sum_{n=0}^\infty \frac{a_n}{(z)_{n+1}},$$0 and $$f(z) = \sum_{n=0}^\infty \frac{a_n}{z(z+1)\cdots(z+n)} = \sum_{n=0}^\infty \frac{a_n}{(z)_{n+1}},$$1, using expansions for the moments involved, translated via Stirling transforms to the inverse factorial basis (Rzadkowski, 2010, Boyadzhiev, 2020).
• Hypergeometric Series and Landau Constants: Inverse factorial expansions have been developed for partial sums of hypergeometric series at unit argument, yielding convergent series whose coefficients depend on parameter excess, with applications to the Landau constants (Paris, 2014).
• Stieltjes Functions and Converging Factors: The converging factor in the remainder of a Stieltjes series admits an explicit inverse factorial expansion, with coefficients recovered recursively from the moment ratio asymptotics. For $$f(z) = \sum_{n=0}^\infty \frac{a_n}{z(z+1)\cdots(z+n)} = \sum_{n=0}^\infty \frac{a_n}{(z)_{n+1}},$$2, the converging factor $$f(z) = \sum_{n=0}^\infty \frac{a_n}{z(z+1)\cdots(z+n)} = \sum_{n=0}^\infty \frac{a_n}{(z)_{n+1}},$$3 has expansion (Borghi, 2024):

$$f(z) = \sum_{n=0}^\infty \frac{a_n}{z(z+1)\cdots(z+n)} = \sum_{n=0}^\infty \frac{a_n}{(z)_{n+1}},$$4

where $$f(z) = \sum_{n=0}^\infty \frac{a_n}{z(z+1)\cdots(z+n)} = \sum_{n=0}^\infty \frac{a_n}{(z)_{n+1}},$$5 is computable by a recursive formula.

3. Connections to Orthogonality, Inversion, and Summation Theory

Inverse factorial expansions anchor a suite of orthogonality and inversion relations in combinatorics and symbolic computation, notably:

• Triangular Transformations: The Stirling numbers of both kinds form lower triangular, mutually inverse matrices, expressing orthogonality relations. For instance,

$$f(z) = \sum_{n=0}^\infty \frac{a_n}{z(z+1)\cdots(z+n)} = \sum_{n=0}^\infty \frac{a_n}{(z)_{n+1}},$$6

(Weniger, 2010, Schreiber, 2023).

• Expansion-Inversion for Polynomial Sequences: For arbitrary Sheffer-type sequences or polynomial bases (central factorials, rising, falling), there are explicit expansions and their inverses in the central factorial basis, with coefficient arrays (central factorial numbers) enjoying rigorous inversion and orthogonality relations (Kim et al., 2022).
• Newton Series and Difference Calculus: Any analytic function (or formal power series) can be expressed in the falling factorial basis ("Newton series"), with the expansion and inversion admissible through explicit finite difference operators, which relate directly to the inverse factorial formalism (Schreiber, 2023).

4. Asymptotics, Convergence, and Error Analysis

The analytic utility of inverse factorial expansions lies in their ability to encode not only convergent representations but full asymptotic expansions, frequently even in settings of factorial divergence:

• Asymptotics of Coefficient Sequences: For power series with coefficients displaying factorial growth (e.g., $$f(z) = \sum_{n=0}^\infty \frac{a_n}{z(z+1)\cdots(z+n)} = \sum_{n=0}^\infty \frac{a_n}{(z)_{n+1}},$$7), the leading behavior and corrections are systematically encoded and computable via the inverse factorial expansion (Borinsky, 2016).
• Convergence Criteria: The convergence of a factorial series $$f(z) = \sum_{n=0}^\infty \frac{a_n}{z(z+1)\cdots(z+n)} = \sum_{n=0}^\infty \frac{a_n}{(z)_{n+1}},$$8 is equivalent to the convergence of a Dirichlet series $$f(z) = \sum_{n=0}^\infty \frac{a_n}{z(z+1)\cdots(z+n)} = \sum_{n=0}^\infty \frac{a_n}{(z)_{n+1}},$$9. This property is crucial in resumming divergent inverse-power or asymptotic series arising in special functions and mathematical physics (Weniger, 2010, Borghi, 2024).
• Error Estimates: Explicit and computable bounds for the error after truncation of an inverse factorial expansion are available, particularly in the context of large-degree special functions (e.g., Jacobi functions), using the structure of the remainder terms and their factorial decay (Nemes, 2024).
• Transformation and Summation of Divergent Series: Inverse factorial series are fundamental in the summation of divergent (typically Stieltjes-type) asymptotic expansions, both in analytic and non-analytic settings. The representation via factorial series regularizes divergent sums, allowing term-by-term summation to yield accurate results where direct summation fails (Weniger, 2010, Borghi, 2024).

5. Structural Extensions: Central and Generalized Factorials

The classical rising and falling factorials admit natural extensions:

• Central Factorial Expansions: Expansions in central factorial bases (e.g., $(z)_k = z(z+1)\cdots(z+k-1)$0) provide finer control over symmetries and appear in connection with orthogonal polynomials and certain special function classes. Expansion–inversion relations and generating functions in this basis generalize the classical theory to broader polynomial families, maintaining explicit mutual inverses and orthogonality relations (Kim et al., 2022).
• General Gamma Ratios and Nørlund–Bernoulli Polynomials: High generality is achieved in expansions involving products and ratios of gamma functions with arbitrary affine arguments, applying Nørlund–Bernoulli and generalized Bernoulli polynomials in the coefficients, yielding both explicit recurrences and determinant forms for these coefficients (Karp et al., 2017).

6. Representative Examples

Below is a table summarizing explicit inverse factorial expansions for several key functions:

Function	Inverse Factorial Expansion	Reference
$(z)_k = z(z+1)\cdots(z+k-1)$1	$(z)_k = z(z+1)\cdots(z+k-1)$2	(Boyadzhiev, 2020)
$(z)_k = z(z+1)\cdots(z+k-1)$3	$(z)_k = z(z+1)\cdots(z+k-1)$4	(Rzadkowski, 2010)
$(z)_k = z(z+1)\cdots(z+k-1)$5	$(z)_k = z(z+1)\cdots(z+k-1)$6	(Weniger, 2010)
Jacobi $(z)_k = z(z+1)\cdots(z+k-1)$7	See theorem in (Nemes, 2024)	(Nemes, 2024)
Polylog $(z)_k = z(z+1)\cdots(z+k-1)$8	$(z)_k = z(z+1)\cdots(z+k-1)$9	(Boyadzhiev, 2020)

7. Applications and Broader Impact

Inverse factorial expansions underpin several analytic and computational methodologies in mathematics and mathematical physics:

• Summation of Divergent Series: Facilitate the summation of formal, factorially divergent series, as in quantum field theory or divergent perturbative treatments, often outperforming traditional resummation techniques (Weniger, 2010, Borghi, 2024).
• Explicit Asymptotics: Enable precise large-degree expansions (e.g., for Jacobi and hypergeometric functions), with computable remainders (Nemes, 2024, Paris, 2014).
• Orthogonality and Polynomial Theory: Structure the theory of factor, central, and rising/falling factorial polynomials, with implications for umbral calculus and combinatorial identities (Schreiber, 2023, Kim et al., 2022).
• Connection to Nonlinear Sequence Transformations: The structure of these expansions is exploited in the design and analysis of powerful nonlinear sequence transformations (e.g., Weniger's transformations) and their convergence acceleration properties for special function computations (Borghi, 2024).

In summary, inverse factorial expansions form a unifying framework in asymptotic analysis, summation theory, combinatorics, and special functions, with both foundational and computational significance.