Multivariate Generalized Lambert Series

• Multivariate generalized Lambert series are extensions of classical Lambert series with several variables or poles, enabling complex analytic and operator-based transformations.
• They employ frameworks such as the λ-derivative and partial theta operators to convert multivariable rational q-kernels into theta products, Appell–Lerch sums, or single Lambert series.
• These series yield explicit transformation identities with applications in modular forms, overpartition ranks, and generalizations of zeta function formulas.

Searching arXiv for recent and foundational papers on multivariate generalized Lambert series. {"query":"all:\"multivariate generalized Lambert series\" OR ti:\"Lambert series\" AND (abs:\"multivariate\" OR abs:\"double Lambert series\")","max_results":10,"sort_by":"submittedDate","sort_order":"descending"} Multivariate generalized Lambert series are extensions of the classical Lambert series

$\sum_{n=1}^{\infty}\frac{a_n q^n}{1-q^n},$

in which multivariateity may enter through several independent pole parameters, several summation indices, or several denominator factors coupled to a single summation variable. In current arXiv literature, the subject includes operator-defined series of the form

$$\sum_{n=0}^{\infty} a_n \prod_{i=1}^{m}\frac{x_i^{\,n+1}}{x_i-\lambda_i^n y_i}\,z^n,$$

double and $k$-fold Lambert series with rational kernels in multiple variables, and doubly infinite $q$-series whose denominators involve several pole parameters $b_1,\dots,b_s$ (López, 20 Jul 2025, Cui et al., 10 Apr 2026, Wei et al., 2018, Chan et al., 2012). Across these formulations, the recurring objective is to convert multivariable rational $q$-kernels into theta products, Appell–Lerch sums, quasimodular forms, or single Lambert series with arithmetic coefficients.

1. Definitions and formal scope

The classical point of departure is the Lambert series

$$L_f(q):=\sum_{n\ge 1}\frac{f(n)\,q^n}{1-q^n}, \qquad |q|<1.$$

Recent work treats generalized forms such as

$$\sum_{n=1}^\infty \frac{a_n\,x\,q^n}{1-x q^n}, \qquad \sum_{n\ge 1}\frac{f(n)\,q^{\alpha n+\beta}}{1-a\,q^{\gamma n+\delta}},$$

and regards multivariateity as arising either from multiple indices or from multiple denominator parameters (Cui et al., 10 Apr 2026, Fang, 5 Apr 2026).

In the double- and $k$-fold setting, a representative template is

$$\sum_{n_1,\dots,n_k\ge 1}A(n_1,\dots,n_k)\prod_{j=1}^k \frac{x_j^{n_j}q^{\alpha_j n_j}}{1-\epsilon_j x_j q^{\beta_j n_j}},$$

with $$\sum_{n=0}^{\infty} a_n \prod_{i=1}^{m}\frac{x_i^{\,n+1}}{x_i-\lambda_i^n y_i}\,z^n,$$0, $$\sum_{n=0}^{\infty} a_n \prod_{i=1}^{m}\frac{x_i^{\,n+1}}{x_i-\lambda_i^n y_i}\,z^n,$$1, and rational or arithmetic weights $$\sum_{n=0}^{\infty} a_n \prod_{i=1}^{m}\frac{x_i^{\,n+1}}{x_i-\lambda_i^n y_i}\,z^n,$$2 (Cui et al., 10 Apr 2026). In the “same-poles” framework of Wei–Zhang, a prototypical multivariate generalized Lambert series is

$$\sum_{n=0}^{\infty} a_n \prod_{i=1}^{m}\frac{x_i^{\,n+1}}{x_i-\lambda_i^n y_i}\,z^n,$$3

with $$\sum_{n=0}^{\infty} a_n \prod_{i=1}^{m}\frac{x_i^{\,n+1}}{x_i-\lambda_i^n y_i}\,z^n,$$4, so that several complex parameters occur simultaneously in the denominator (Wei et al., 2018).

A distinct but compatible formalism is the operator-based construction of multivariate generalized Lambert series: $$\sum_{n=0}^{\infty} a_n \prod_{i=1}^{m}\frac{x_i^{\,n+1}}{x_i-\lambda_i^n y_i}\,z^n,$$5 which packages many classical and nonclassical Lambert-type series into a single transformation theory (López, 20 Jul 2025). The coexistence of these definitions is not merely terminological variation. It reflects different analytic agendas: contour or theta-product identities in the same-poles setting, systematic series rearrangement in the multiple-index setting, and operator-calculus transforms in the $$\sum_{n=0}^{\infty} a_n \prod_{i=1}^{m}\frac{x_i^{\,n+1}}{x_i-\lambda_i^n y_i}\,z^n,$$6-derivative setting.

2. Operator-theoretic constructions and transformation formulas

A systematic operator framework is developed through the $$\sum_{n=0}^{\infty} a_n \prod_{i=1}^{m}\frac{x_i^{\,n+1}}{x_i-\lambda_i^n y_i}\,z^n,$$7-derivative

$$\sum_{n=0}^{\infty} a_n \prod_{i=1}^{m}\frac{x_i^{\,n+1}}{x_i-\lambda_i^n y_i}\,z^n,$$8

and the Partial Theta operator

$$\sum_{n=0}^{\infty} a_n \prod_{i=1}^{m}\frac{x_i^{\,n+1}}{x_i-\lambda_i^n y_i}\,z^n,$$9

Its key action on monomials is

$k$0

which converts operator calculus directly into generalized Lambert kernels (López, 20 Jul 2025).

If $k$1, the base transformation formula is

$k$2

This identity recovers the classical Lambert series by the specialization $k$3 with $k$4, giving

$k$5

The same formalism yields generalized Lambert–Mehler and Lambert–Rogers type series, including

$k$6

and it extends to bivariate dilated series involving a second scaling parameter $k$7 (López, 20 Jul 2025).

The multivariate theorem is

$k$8

In the symmetric case $k$9, $q$0, and $q$1, this becomes

$q$2

Analytically, the transformed sums are controlled by geometric decay in the ratios $q$3, while singularities occur at the simple poles $q$4 (López, 20 Jul 2025).

3. Double Lambert series and systematic rearrangement

A second major strand studies genuinely multivariate Lambert series with two summation indices. Cui and Tang consider double Lambert series such as

$q$5

together with auxiliary series

$q$6

$q$7

and

$q$8

Their method consists of geometric and alternating geometric expansions, change of variables, and justified interchange of summations under $q$9, culminating in the explicit representation

$b_1,\dots,b_s$0

from which $b_1,\dots,b_s$1 follows immediately (Cui et al., 10 Apr 2026).

Fang studies the same ADS–Y double Lambert series in the equivalent form

$b_1,\dots,b_s$2

and proves the parity conjecture by decomposing

$b_1,\dots,b_s$3

showing $b_1,\dots,b_s$4, and factorizing

$b_1,\dots,b_s$5

The resulting $b_1,\dots,b_s$6 structure isolates odd powers and makes the parity argument transparent (Fang, 5 Apr 2026).

A complementary arithmetic application is the Amdeberhan–Andrews–Ballantine conjecture on the double Lambert series

$b_1,\dots,b_s$7

The proved identity is

$b_1,\dots,b_s$8

and its base form collapses the double series to a single Lambert series: $b_1,\dots,b_s$9 This yields the representation

$q$0

linking a double generalized Lambert series directly to the quasi-modular form $q$1 (Kumar et al., 20 May 2026).

4. Same-pole identities and the $q$2-series mechanism

In the Wei–Zhang framework, multivariate generalized Lambert series are organized by their common pole sets. The auxiliary $q$3-series is defined by

$q$4

and satisfies identities such as $q$5, $q$6, $q$7, and $q$8. It mediates between generalized Lambert series and theta-product quotients through

$q$9

where $$L_f(q):=\sum_{n\ge 1}\frac{f(n)\,q^n}{1-q^n}, \qquad |q|<1.$$0 (Wei et al., 2018).

The central structural result is the existence of two master identities, one for double poles and one for single poles. Their defining feature is decoupling: the numerator parameters $$L_f(q):=\sum_{n\ge 1}\frac{f(n)\,q^n}{1-q^n}, \qquad |q|<1.$$1 are separated from the pole parameters $$L_f(q):=\sum_{n\ge 1}\frac{f(n)\,q^n}{1-q^n}, \qquad |q|<1.$$2, and the right-hand sides become linear combinations of generalized Lambert series with the same poles $$L_f(q):=\sum_{n\ge 1}\frac{f(n)\,q^n}{1-q^n}, \qquad |q|<1.$$3. This same-poles formalism is designed to combine or separate multivariate generalized Lambert series whose denominators differ only by parameter placement or multiplicity. It is especially effective when $$L_f(q):=\sum_{n\ge 1}\frac{f(n)\,q^n}{1-q^n}, \qquad |q|<1.$$4 is a unit root, because then double poles arise naturally.

The same framework is used to study $$L_f(q):=\sum_{n\ge 1}\frac{f(n)\,q^n}{1-q^n}, \qquad |q|<1.$$5-dissections of rank generating functions for overpartitions modulo $$L_f(q):=\sum_{n\ge 1}\frac{f(n)\,q^n}{1-q^n}, \qquad |q|<1.$$6. In that setting, generalized Lambert series with poles at $$L_f(q):=\sum_{n\ge 1}\frac{f(n)\,q^n}{1-q^n}, \qquad |q|<1.$$7 appear, and the master identities produce compact theta-product combinations as well as formulas involving both $$L_f(q):=\sum_{n\ge 1}\frac{f(n)\,q^n}{1-q^n}, \qquad |q|<1.$$8 and $$L_f(q):=\sum_{n\ge 1}\frac{f(n)\,q^n}{1-q^n}, \qquad |q|<1.$$9. The paper also relates these rank-difference formulas to the third-order mock theta functions $$\sum_{n=1}^\infty \frac{a_n\,x\,q^n}{1-x q^n}, \qquad \sum_{n\ge 1}\frac{f(n)\,q^{\alpha n+\beta}}{1-a\,q^{\gamma n+\delta}},$$0 and $$\sum_{n=1}^\infty \frac{a_n\,x\,q^n}{1-x q^n}, \qquad \sum_{n\ge 1}\frac{f(n)\,q^{\alpha n+\beta}}{1-a\,q^{\gamma n+\delta}},$$1, so the multivariate generalized Lambert series act as the analytic intermediary between overpartition combinatorics and mock-modular objects (Wei et al., 2018).

5. Appell–Lerch sums and Rank–Crank type PDEs

Chan’s multivariate generalized Lambert series identity provides a unifying source for Watson’s and Jackson’s classical generalized Lambert series identities. In the formulation used by Chan, Dixit, and Garvan, the common target is the higher-order Appell–Lerch function

$$\sum_{n=1}^\infty \frac{a_n\,x\,q^n}{1-x q^n}, \qquad \sum_{n\ge 1}\frac{f(n)\,q^{\alpha n+\beta}}{1-a\,q^{\gamma n+\delta}},$$2

together with the $$\sum_{n=1}^\infty \frac{a_n\,x\,q^n}{1-x q^n}, \qquad \sum_{n\ge 1}\frac{f(n)\,q^{\alpha n+\beta}}{1-a\,q^{\gamma n+\delta}},$$3-rank generating series $$\sum_{n=1}^\infty \frac{a_n\,x\,q^n}{1-x q^n}, \qquad \sum_{n\ge 1}\frac{f(n)\,q^{\alpha n+\beta}}{1-a\,q^{\gamma n+\delta}},$$4. Their theorem gives the decomposition

$$\sum_{n=1}^\infty \frac{a_n\,x\,q^n}{1-x q^n}, \qquad \sum_{n\ge 1}\frac{f(n)\,q^{\alpha n+\beta}}{1-a\,q^{\gamma n+\delta}},$$5

exhibiting the $$\sum_{n=1}^\infty \frac{a_n\,x\,q^n}{1-x q^n}, \qquad \sum_{n\ge 1}\frac{f(n)\,q^{\alpha n+\beta}}{1-a\,q^{\gamma n+\delta}},$$6-rank series as an Appell–Lerch term plus explicit theta corrections (Chan et al., 2012).

The multivariate mechanism enters through a specialization $$\sum_{n=1}^\infty \frac{a_n\,x\,q^n}{1-x q^n}, \qquad \sum_{n\ge 1}\frac{f(n)\,q^{\alpha n+\beta}}{1-a\,q^{\gamma n+\delta}},$$7 in Chan’s identity, followed by $$\sum_{n=1}^\infty \frac{a_n\,x\,q^n}{1-x q^n}, \qquad \sum_{n\ge 1}\frac{f(n)\,q^{\alpha n+\beta}}{1-a\,q^{\gamma n+\delta}},$$8-differentiation. This produces differential identities for $$\sum_{n=1}^\infty \frac{a_n\,x\,q^n}{1-x q^n}, \qquad \sum_{n\ge 1}\frac{f(n)\,q^{\alpha n+\beta}}{1-a\,q^{\gamma n+\delta}},$$9 with quasimodular coefficients. In particular, Corollary 4.5 states that there exist quasimodular forms $k$0 such that

$k$1

This realizes higher-order Rank–Crank type PDEs as direct consequences of multivariate generalized Lambert series identities.

The classical Atkin–Garvan Rank–Crank PDE is the case $k$2: $k$3 For $k$4, the same mechanism yields the level $k$5 Appell-function identity

$k$6

Section 2 of the paper further shows, by extending the Atkin–Swinnerton-Dyer elliptic-function method, that the relevant Lambert-series combinations are entire elliptic functions in the auxiliary parameters and therefore constant. That elliptic-function argument explains why the multivariate identities collapse to parameter-independent product expressions (Chan et al., 2012).

6. Higher-power analogues, zeta values, and proposed multivariate extensions

A different but closely related direction arises from the generalized Lambert series

$k$7

or, with $k$8,

$k$9

This series is not multivariate in summation index, but it develops a transformation machinery—based on Mellin inversion, contour shifts, Hurwitz’s formula, discrete trigonometric expansions, and sums of Raabe integrals—that explicitly suggests a multivariate generalization (Dixit et al., 2018).

The paper derives transformation formulas that connect $$\sum_{n_1,\dots,n_k\ge 1}A(n_1,\dots,n_k)\prod_{j=1}^k \frac{x_j^{n_j}q^{\alpha_j n_j}}{1-\epsilon_j x_j q^{\beta_j n_j}},$$0 to Hurwitz zeta values, Bernoulli polynomials, digamma functions, logarithms, and odd zeta values. Among the consequences are two-parameter generalizations of Ramanujan’s formula for $$\sum_{n_1,\dots,n_k\ge 1}A(n_1,\dots,n_k)\prod_{j=1}^k \frac{x_j^{n_j}q^{\alpha_j n_j}}{1-\epsilon_j x_j q^{\beta_j n_j}},$$1, a two-parameter generalization of the transformation formula for $$\sum_{n_1,\dots,n_k\ge 1}A(n_1,\dots,n_k)\prod_{j=1}^k \frac{x_j^{n_j}q^{\alpha_j n_j}}{1-\epsilon_j x_j q^{\beta_j n_j}},$$2, identities relating $$\sum_{n_1,\dots,n_k\ge 1}A(n_1,\dots,n_k)\prod_{j=1}^k \frac{x_j^{n_j}q^{\alpha_j n_j}}{1-\epsilon_j x_j q^{\beta_j n_j}},$$3 for odd $$\sum_{n_1,\dots,n_k\ge 1}A(n_1,\dots,n_k)\prod_{j=1}^k \frac{x_j^{n_j}q^{\alpha_j n_j}}{1-\epsilon_j x_j q^{\beta_j n_j}},$$4, and transcendence or irrationality criteria of Zudilin- and Rivoal-type.

The explicitly proposed multivariate analogue is

$$\sum_{n_1,\dots,n_k\ge 1}A(n_1,\dots,n_k)\prod_{j=1}^k \frac{x_j^{n_j}q^{\alpha_j n_j}}{1-\epsilon_j x_j q^{\beta_j n_j}},$$5

where $$\sum_{n_1,\dots,n_k\ge 1}A(n_1,\dots,n_k)\prod_{j=1}^k \frac{x_j^{n_j}q^{\alpha_j n_j}}{1-\epsilon_j x_j q^{\beta_j n_j}},$$6 is a polynomial, or a product of powers $$\sum_{n_1,\dots,n_k\ge 1}A(n_1,\dots,n_k)\prod_{j=1}^k \frac{x_j^{n_j}q^{\alpha_j n_j}}{1-\epsilon_j x_j q^{\beta_j n_j}},$$7, and $$\sum_{n_1,\dots,n_k\ge 1}A(n_1,\dots,n_k)\prod_{j=1}^k \frac{x_j^{n_j}q^{\alpha_j n_j}}{1-\epsilon_j x_j q^{\beta_j n_j}},$$8, $$\sum_{n_1,\dots,n_k\ge 1}A(n_1,\dots,n_k)\prod_{j=1}^k \frac{x_j^{n_j}q^{\alpha_j n_j}}{1-\epsilon_j x_j q^{\beta_j n_j}},$$9. The envisioned method is to apply Mellin transforms in each variable, use multidimensional variants of Hurwitz’s formula and finite Fourier expansions, and introduce multivariate Raabe-type integrals. The paper further envisages modular-like transformations coupling $$\sum_{n=0}^{\infty} a_n \prod_{i=1}^{m}\frac{x_i^{\,n+1}}{x_i-\lambda_i^n y_i}\,z^n,$$00 through

$$\sum_{n=0}^{\infty} a_n \prod_{i=1}^{m}\frac{x_i^{\,n+1}}{x_i-\lambda_i^n y_i}\,z^n,$$01

together with relations among multivariate odd zeta combinations and Lambert-type multisums. The stated obstacles are convergence, pole resonance across several Mellin variables, and the need for a multivariate Poisson or Guinand summation adapted to mixed trigonometric kernels (Dixit et al., 2018).

Taken together, these developments show that multivariate generalized Lambert series form not a single rigid class but a family of closely related analytic constructions. Their common structure is the controlled interaction of rational $$\sum_{n=0}^{\infty} a_n \prod_{i=1}^{m}\frac{x_i^{\,n+1}}{x_i-\lambda_i^n y_i}\,z^n,$$02-kernels, parameter symmetries, and transformation theory, with consequences spanning parity phenomena, theta-product identities, quasimodular forms, overpartition ranks, higher Appell functions, odd zeta values, and prospective multidimensional extensions.