Infinite Series Sum Transformation

• Infinite series sum transformation is the process of converting complex infinite sums into alternative forms such as product, integral, or generating function representations.
• Techniques apply to $q$-series, hypergeometric functions, and combinatorial identities, facilitating theoretical analysis and efficient numerical computation.
• These transformations are central to analytic continuation, accelerated convergence methods, and modern computational frameworks in mathematics.

An infinite series sum transformation is any method or formula that converts, rearranges, or evaluates infinite sums—typically of arithmetic, combinatorial, or special function sequences—into alternative forms that are more tractable for theoretical analysis, closed-form evaluation, or efficient numerical calculation. These transformations constitute one of the central techniques in analytic combinatorics, special function theory, analytic number theory, and computational mathematics, providing connections between series expansions, integral representations, generating functions, and algebraic identities. Below, key aspects of infinite series sum transformation are presented, drawing from core results in basic hypergeometric series, automorphic forms, analytic continuation, summability calculus, probability theory, and computational methods.

1. Formalism and Notational Conventions

Infinite series transformations operate on formal sums of the form

$S = \sum_{k=0}^\infty f(k),$

where $f: \mathbb{N} \to \mathbb{C}$ is the sequence under consideration. In the context of $q$-series and basic hypergeometric series, summands typically involve $q$-shifted factorials, i.e.,

$$(x; q)_n = \prod_{j=0}^{n-1} (1 - xq^j),\qquad (x_1,\ldots,x_r;q)_n = \prod_{j=1}^r (x_j;q)_n,$$

and the series may be unilateral, bilateral, or of higher depth (multi-sum).

Transformations may map a sum to:

• A different (but equivalent) infinite sum, often with alternative parameters or in terms of different basic hypergeometric series.
• An infinite product, recognizing underlying product structures via the identities such as Euler’s pentagonal number theorem or Rogers-Ramanujan identities (Laughlin, 2019, Xu et al., 2 Mar 2024).
• An integral representation, as via Laplace or Mellin transforms (Stenlund, 2014).
• A generating function, relating series coefficients to function values or derivatives (Schmidt, 2016).
• An analytic continuation, generalizing sums to non-integer or complex upper and lower bounds (Saleh, 2 Feb 2024).

The full range of available transformations is underpinned by analytic, algebraic, and combinatorial properties, and crucially, by the convergence properties of the series.

2. Transformation Formulas in Basic Hypergeometric Series

A principal domain of series transformations is the theory of basic hypergeometric series (or $q$-series), which generalizes ordinary power and hypergeometric series. The canonical form for a bilateral basic hypergeometric series is

$${}_r\psi_s\left[\begin{array}{c} a_1,\ldots,a_r \ b_1,\ldots,b_s \end{array}; q, z \right] = \sum_{k=-\infty}^{\infty} \frac{(a_1,\ldots,a_r;q)_k}{(b_1,\ldots,b_s;q)_k} z^k,$$

subject to convergence constraints.

Key families of transformations include:

• Six-Variable Generalizations: As in "Several transformation formulas for basic hypergeometric series" (Wei et al., 2013), identities are established that generalize Andrews’ four-variable extension of Ramanujan's ${}_1\psi_1$ sum to six variables, substantially increasing parametric flexibility. A representative form is:

$$\sum_{k=0}^{\infty} \frac{(-q/a, AB/ab; q)_k\,(-b)^k\,(x, y, -xa/A, -xa/B; q)_k\,(1-q^{2k}x)}{(q, -xa, xq/y; q)_k} \cdots = {}_8W_7\left(\frac{xa^2}{yA}; \cdots; q, -bq/A \right)$$

The importance of these formulas lies in their ability to transform, specialize, or "split" bilateral sums, facilitating the evaluation or rearrangement of complicated $q$-series via unilateral series or $q$-products.

• Generalizations of Bailey’s ${}_3\psi_3$ Summation: Extending well-known summation identities using new limiting relations (see Lemmas 5 and 6 in (Wei et al., 2013)), e.g.,

$$\sum_{k=-\infty}^\infty \frac{(b, c, d; q)_k}{(q/b, q/c, q/d; q)_k} \left( \frac{q}{bcd} \right)^k = \frac{(q, q/bc, q/bd, q/cd; q)_\infty}{(q/b, q/c, q/d, q/bcd; q)_\infty}$$

These generalized summations allow for transformations between bilateral and unilateral series, crucial for the paper of partition identities, orthogonal polynomials expressible in $q$-series, and $q$-supercongruences.

• Multi-sum to Single-sum Reductions: As in (Laughlin, 2019), general theorems reduce $k$-fold nested sums to single sums multiplied by infinite $q$-products; e.g., the transformation

$$\sum_{m_1 \geq m_2 \geq \dots \geq m_{k+1} \geq 0} \cdots = \prod \left(\text{q-products}\right) \cdot \sum_{n = 0}^\infty g(n)$$

enables the explicit identification of product structures underlying sums of combinatorial and partition-theoretic significance.

• Pfaff–Saalschütz/Triple Series Transformations: The recent developments in (Xu et al., 2 Mar 2024) and (Liu, 25 Jan 2024) exhibit general $q$-series transformations involving arbitrary parameter sequences, allowing, for instance, efficient derivation of multi-sum Rogers–Ramanujan–Slater identities and generating function identities for $q$-orthogonal polynomials.

3. Analytic, Integral, and Generating Function Transformations

Infinite series can often be recast as integrals or generating functions through methods including:

• Laplace Transform Techniques: As elucidated in (Stenlund, 2014), if $g(k)$ admits an inverse Laplace transform $G(t)$,

$g(k) = \int_0^\infty e^{-kt} G(t)\,dt,$

then

$$S = \sum_{k=1}^\infty g(k) = \int_0^\infty G(t) \frac{e^{-t}}{1 - e^{-t}}\,dt.$$

This reduces the summation problem to an integration problem, which is sometimes tractable using special function identities or further transforms.

• Generating Function Transforms: A new class of transformations (Schmidt, 2016) rewrites sums

$$\sum_{n=1}^\infty \frac{g_n}{n^k} z^n = \sum_{j=1}^\infty S^*(k+2, j) z^j G^{(j)}(z),$$

where $S^*(k+2, j)$ are generalized coefficients related to harmonic numbers and $G(z)$ is the ordinary generating function of the sequence $\{g_n\}$. This approach yields generalized polylogarithm expansions and provides "inverse" relations to the traditional generating function power sum formulas.

4. Connections to Probability, Algebra, and Analytic Continuation

Series transformations emerge naturally outside classical special function theory:

• Probability and Union Estimates: (Grahl et al., 2016) develops a transformation between a sequence of probabilities $\{x_n\}$ (of independent events) and the probability of their union via

$$T_n = x_n \prod_{k=1}^{n-1}(1 - x_k),\qquad \sum_{n=1}^\infty T_n = 1 - \prod_{n=1}^\infty (1 - x_n).$$

This logsums the effect of overlap, translating between additive and multiplicative summation—a classic transformation theme.

• Analytic Continuation and Summability Calculus: The generalization of finite sums to non-integer bounds (Saleh, 2 Feb 2024) via

$$\sum_{k=1}^x f(k) = xL + \sum_{k=1}^\infty \big(f(k) - f(k+x)\big),$$

where $L = \lim_{k\to\infty} f(k)$, provides a method for analytic continuation, approximate evaluation, and explicit error analysis of truncated sums.

• Automorphic and Modular Transformations: Modular transformation formulas for Eisenstein series, Lambert series, and Dedekind sums (Dixit et al., 2020, Dağlı et al., 2015) relate double or higher-depth series to each other under the action of $\mathrm{SL}_2(\mathbb{Z})$, introducing additional terms involving periodic Bernoulli functions or Bessel integrals which account for modular obstructions and reciprocity relations.

5. Special Functions, Quantum Models, and Orthogonal Expansions

Transformations also arise by exploiting orthogonality or underlying physical/functional models:

• Eigenfunction Expansions and Quantum Problems: Quantum models with well-understood Hilbert space bases, such as the particle in an infinite potential well, yield closed-form summation formulas for series involving Bessel functions and hypergeometric functions by expanding "test" states and applying normalization (Mali et al., 2023). For instance,

$$\sum_{k=0}^\infty \frac{J_1^2((2k+1)\pi/2)}{(2k+1)^2} = \frac{1}{3},$$

is directly encoded by normalization in quantum mechanics.

• Binomial and Pochhammer Sums, Cyclotomic Polylogarithms: Infinite series with combinatorial and binomial structure (e.g., sums over $(2n \choose n)^{-1}$ or nested sums with Pochhammer symbols) are reducible to combinations of iterated integrals or cyclotomic harmonic polylogarithms, which are then expressible in terms of $\pi$, $\zeta$-values, and logarithms via algebraic reduction (Ablinger, 2015, Ablinger, 2019).

6. Acceleration of Convergence and Computational Methods

Slowly converging or divergent series are ubiquitously transformed for accelerated evaluation:

• Nonlinear Convergence Acceleration (d-Transformation): (Levin, 2 Aug 2025, Sidi, 2017) present a general framework for accelerating series with known or inferable asymptotic expansions for the remainder. The d-transformation constructs a system

$$S \simeq A_N(f) + \sum_{k=0}^{m-1} \Delta^k f(N) N^k \sum_{i=0}^{r-1} \beta_{k,i} N^{-i},$$

and solves for $S$ by matching values at shifted indices, systematically eliminating leading error orders and yielding rapid, sometimes exponential, convergence even in cases with divergent original series. The method is particularly powerful for series whose terms are solutions to linear difference equations (sequences in the class $B^{(m)}$).

• Implementation and Stability Considerations: Advanced algorithms (e.g., the recursive W-algorithm for $\tilde{d}^{(m)}$ (Sidi, 2017)) are used to recursively compute improved approximations and error indicators, adapting sampling choices and ensuring stability under floating-point arithmetic constraints.

7. Structural and Algebraic Perspectives

• Formal Bernoulli and Generating Series: (Alekseyev et al., 6 Jun 2025) generalizes Ramanujan's formulas for half-integer power sums by lifting Bernoulli polynomials to formal series

$$B_{(\alpha)}(z) = \sum_{i=0}^\infty \binom{\alpha}{i} B_i z^i,$$

leading to exact representations of sums such as

$$\sum_{i=1}^{n} i^{k/2} = C_k + \sqrt{n} P_k(n) + \sum_{i\,\mathrm{odd}}^{k+2} A_i^k \tau(n, i),$$

where $P_k(n)$ is a polynomial and $\tau(n, i)$ an explicitly parameterized infinite series.

• Unified Metric Approach: (Losonczi, 2021) employs the Hausdorff metric to define sum extensions, showing that classical notions of unordered sum, integral, and arc length are unified as the continuous extension of a sum operator from finite subsets to a full index set.
• Transcendence via Diophantine Approximation: Series specifically constructed to be Liouville numbers can be shown, via Roth's theorem, to yield transcendental values upon summation, combination, or even quotients, as long as rational approximations converge sufficiently rapidly (Ahallal et al., 21 May 2024).

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In conclusion, infinite series sum transformations encompass a multifaceted toolkit unifying analytic, combinatorial, algebraic, and computational perspectives. They are central to the derivation and simplification of summation identities, modular and $q$-series relations, analytic continuations, and high-precision numerical evaluations, supporting a broad spectrum of inquiry in modern mathematics and its computational applications.