---
title: Multivariate Generalized Lambert Series
url: https://www.emergentmind.com/topics/multivariate-generalized-lambert-series
type: topic
---

# Multivariate Generalized Lambert Series

Searching arXiv for recent and foundational papers on multivariate generalized Lambert series.
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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 \(b_1,\dots,b_s\) [2507.15160] [2604.08839] [1801.04643] [1201.1663]. 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 [2604.08839] [2604.06242].

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 \(\alpha_j,\beta_j\in\mathbb{Z}_{\ge 0}\), \(\epsilon_j\in\{\pm1\}\), and rational or arithmetic weights \(A(n_1,\dots,n_k)\) [2604.08839]. In the “same-poles” framework of Wei–Zhang, a prototypical multivariate generalized Lambert series is
\[
L(b_1,\dots,b_s;q):=\sum_{n=-\infty}^{\infty}
(-1)^{\ell n}q^{\ell n(n+1)/2}\prod_{j=1}^s (1-b_j q^n)^{-\alpha_j},
\]
with \(\alpha_j\in\{1,2\}\), so that several complex parameters occur simultaneously in the denominator [1801.04643].

A distinct but compatible formalism is the operator-based construction of multivariate generalized Lambert series:
\[
\sum_{n=0}^{\infty}a_n\prod_{i=1}^{m}\frac{x_i^{\,n+1}}{x_i-\lambda_i^n y_i}\,z^n,
\qquad x_i\neq \lambda_i^n y_i,
\]
which packages many classical and nonclassical Lambert-type series into a single transformation theory [2507.15160]. 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 \(\lambda\)-derivative setting.

## 2. Operator-theoretic constructions and transformation formulas

A systematic operator framework is developed through the \(\lambda\)-derivative
\[
(\mathbf{D}_{\lambda}f)(x)=
\begin{cases}
\dfrac{f(\lambda x)}{x}, & x\neq 0,\\[6pt]
\lim_{x\to 0}\dfrac{f(\lambda x)}{x}, & x=0,
\end{cases}
\]
and the Partial Theta operator
\[
\theta(y\mathbf{D}_{\lambda})
:=\sum_{n=0}^{\infty}\lambda^{\binom{n}{2}}y^{n}\,\mathbf{D}_{\lambda}^{n}.
\]
Its key action on monomials is
\[
\theta(y\mathbf{D}_{\lambda})\{x^{n}\}
=
\frac{x^{n+1}}{x-\lambda^{n}y},
\]
which converts operator calculus directly into generalized Lambert kernels [2507.15160].

If \(f(x)=\sum_{n=0}^{\infty}a_n x^n\), the base transformation formula is
\[
\sum_{n=0}^{\infty}a_n\,\frac{x^{\,n+1}}{x-\lambda^{n}y}\,z^{n}
=
\sum_{k=0}^{\infty}\left(\frac{y}{x}\right)^k f(\lambda^k xz).
\]
This identity recovers the classical Lambert series by the specialization \((x,y,z,\lambda)=(1,1,q,q)\) with \(a_0=0\), giving
\[
\sum_{n=1}^{\infty}a_n\frac{q^n}{1-q^n}.
\]
The same formalism yields generalized Lambert–Mehler and Lambert–Rogers type series, including
\[
\sum_{n,m\ge 0}a_n b_m\,
\frac{x^{\,n+m+1}}{x-\lambda^{n+m}y}\,t^n s^m
=
\sum_{k=0}^{\infty}\left(\frac{y}{x}\right)^k
f(\lambda^k tx)\,g(\lambda^k sx),
\]
and it extends to bivariate dilated series involving a second scaling parameter \(\mu\) [2507.15160].

The multivariate theorem is
\[
\sum_{n=0}^{\infty}a_n\prod_{i=1}^{m}
\frac{x_i^{\,n+1}}{x_i-\lambda_i^{n}y_i}\,z^n
=
\sum_{k_1,\ldots,k_m\ge 0}
\prod_{i=1}^{m}\left(\frac{y_i}{x_i}\right)^{k_i}
f\!\left(\lambda_1^{k_1}\cdots \lambda_m^{k_m}x_1\cdots x_m z\right).
\]
In the symmetric case \(x_1=\cdots=x_m=x\), \(y_1=\cdots=y_m=y\), and \(\lambda_1=\cdots=\lambda_m=\lambda\), this becomes
\[
\sum_{n=0}^{\infty}a_n\left(\frac{x^{\,n+1}}{x-\lambda^n y}\right)^m z^n
=
\sum_{k_1,\ldots,k_m\ge 0}
\left(\frac{y}{x}\right)^{k_1+\cdots+k_m}
f\!\left(\lambda^{k_1+\cdots+k_m}x^m z\right).
\]
Analytically, the transformed sums are controlled by geometric decay in the ratios \(y_i/x_i\), while singularities occur at the simple poles \(x_i=\lambda_i^n y_i\) [2507.15160].

## 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
\[
Y(q):=\sum_{n=1}^\infty\sum_{m=1}^\infty
\frac{(-1)^m q^{2nm+m}}{(1-q^{2m-1})(1+q^n)},
\]
together with auxiliary series
\[
X(q):=\sum_{k=1}^\infty\sum_{n=1}^\infty
\frac{(-1)^k q^{2kn+k}}{(1-q^n)(1-q^{2k-1})},
\]
\[
Z(q):=\sum_{k=1}^\infty\sum_{n=1}^\infty
\frac{q^{2kn}}{(1+q^{2n-1})(1-q^{2k-1})},
\]
and
\[
A(q):=\sum_{k=1}^\infty\sum_{n=k}^\infty
\frac{q^{k+n}}{(1+q^n)(1+q^{2k-1})}.
\]
Their method consists of geometric and alternating geometric expansions, change of variables, and justified interchange of summations under \(|q|<1\), culminating in the explicit representation
\[
Y(q)=-q\,\frac{(q^4;q^4)_\infty^4}{(q^2;q^2)_\infty^2}
\sum_{k=1}^{\infty}\frac{q^{2k}}{1+q^{2k}},
\]
from which \(Y(-q)=-Y(q)\) follows immediately [2604.08839].

Fang studies the same ADS–Y double Lambert series in the equivalent form
\[
Y(q)=\sum_{m=1}^\infty \frac{(-1)^m q^m}{1-q^{2m-1}}
\sum_{k=2m}^\infty \frac{(-1)^k q^k}{1-q^k},
\]
and proves the parity conjecture by decomposing
\[
Y(q)=D_1(q)-D_2(q)-A(q)+B_1(q),
\]
showing \(B_1(q)=A(-q)\), and factorizing
\[
D_1(q)-D_2(q)
=
q\,\frac{(q^4;q^4)_\infty^4}{(q^2;q^2)_\infty^2}
\sum_{k\ge 1}\frac{(-1)^{k-1}q^{2k}}{1-q^{2k}}.
\]
The resulting \(q\times F(q^2)\) structure isolates odd powers and makes the parity argument transparent [2604.06242].

A complementary arithmetic application is the Amdeberhan–Andrews–Ballantine conjecture on the double Lambert series
\[
\sum_{m,k\ge1}\frac{q^{mk2^a}}
{(1+q^{k2^{a-1}})(1-q^{2m-1})}.
\]
The proved identity is
\[
[q^{N2^a}]
\sum_{m,k\ge1}\frac{q^{mk2^a}}
{(1+q^{k2^{a-1}})(1-q^{2m-1})}
=
\sigma_1(N),
\]
and its base form collapses the double series to a single Lambert series:
\[
\sum_{m,k\ge1}\frac{q^{2mk}}{(1+q^k)(1-q^{2m-1})}
=
\sum_{n\ge1}\frac{q^{2n}}{(1-q^{2n})^2}
=
\sum_{N\ge1}\sigma_1(N)q^{2N}.
\]
This yields the representation
\[
E_2(q)=1-24\sum_{m,k\ge1}\frac{q^{2mk}}{(1+q^k)(1-q^{2m-1})},
\]
linking a double generalized Lambert series directly to the quasi-modular form \(E_2\) [2605.21163].

## 4. Same-pole identities and the \(S\)-series mechanism

In the Wei–Zhang framework, multivariate generalized Lambert series are organized by their common pole sets. The auxiliary \(S\)-series is defined by
\[
S(a_1,\dots,a_r;q)
:=
\sum_{u=1}^r\sum_{n=0}^{\infty}
\left(
\frac{1}{1-a_u q^n}
-
\frac{1}{1-a_u^{-1} q^{n+1}}
\right),
\]
and satisfies identities such as \(S(aq)=S(a)+1\), \(S(q/a)=-S(a)\), \(S(-1)=-1/2\), and \(S(-q)=1/2\). It mediates between generalized Lambert series and theta-product quotients through
\[
P(a,b,c;q):=
\frac{[ab,bc,ca]_\infty (q;q)_\infty^2}{[a,b,c,abc]_\infty}
=
1+S(a)+S(b)+S(c)-S(abc),
\]
where \([a]_\infty=(a;q)_\infty(q/a;q)_\infty\) [1801.04643].

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 \(a_i\) are separated from the pole parameters \(b_j\), and the right-hand sides become linear combinations of generalized Lambert series with the same poles \(1-b_j q^n=0\). 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 \(-1\) is a unit root, because then double poles arise naturally.

The same framework is used to study \(3\)-dissections of rank generating functions for overpartitions modulo \(6\). In that setting, generalized Lambert series with poles at \(1\pm q^{3n+1}\) appear, and the master identities produce compact theta-product combinations as well as formulas involving both \((1+q^{3n+1})^{-1}\) and \((1+q^{3n+1})^{-2}\). The paper also relates these rank-difference formulas to the third-order mock theta functions \(\omega(q)\) and \(\rho(q)\), so the multivariate generalized Lambert series act as the analytic intermediary between overpartition combinatorics and mock-modular objects [1801.04643].

## 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
\[
A_{2k-1}(z,q)
:=
\sum_{n=-\infty}^{\infty}
(-1)^n q^{(2k-1)n(n+1)/2}\frac{1}{1-zq^n},
\]
together with the \(k\)-rank generating series \(R_k(z,q)\). Their theorem gives the decomposition
\[
\frac{R_k(z,q)}{(q)_\infty}
=
z^{k-1}(1-z)A_{2k-1}(z,q)
-z^{-1}\vartheta_{1,2k-1}(q)
+z(1-z)\sum_{m=0}^{k-3}z^m\vartheta_{2m+3,2k-1}(q),
\]
exhibiting the \(k\)-rank series as an Appell–Lerch term plus explicit theta corrections [1201.1663].

The multivariate mechanism enters through a specialization \(x_i=\xi^i\) in Chan’s identity, followed by \(\xi\)-differentiation. This produces differential identities for \(A_{2m+1}(z,q)\) with quasimodular coefficients. In particular, Corollary 4.5 states that there exist quasimodular forms \(f_j(q)\) such that
\[
H_{2m+1}^{m} A_{2m+1}(z,q)
+\sum_{k=1}^{m} f_{m-k}(q)\,
H_{2m+1}^{m-k} A_{2m+1}(z,q)
=
(2m)!\,[C^*(z,q)]^{2m+1}(q)_\infty^{2m+1}.
\]
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 \(m=1\):
\[
z (q)_\infty^2 [C^*(z,q)]^3
=
(3\theta_q + 2\theta_z + \theta_z^2) R^*(z,q).
\]
For \(m=2\), the same mechanism yields the level \(5\) Appell-function identity
\[
24 (q)_\infty^2 [C^*(z,q)]^5
=
(H_1^2 - E_4(q)) G^{(5)}(z,q).
\]
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 [1201.1663].

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

A different but closely related direction arises from the generalized Lambert series
\[
S_{N,h}(a,x)
:=
\sum_{n=1}^{\infty}
\frac{n^{\,N-2h} e^{-a n^{N} x}}{1-e^{-n^{N}x}},
\qquad
0<a\le 1,\ x>0,\ N\in\mathbb{N},\ h\in\mathbb{Z},
\]
or, with \(q=e^{-x}\),
\[
S_{N,h}(a;q)
=
\sum_{n=1}^{\infty}
\frac{n^{\,N-2h} q^{a n^N}}{1-q^{n^N}}.
\]
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 [1801.09181].

The paper derives transformation formulas that connect \(S_{N,h}(a,x)\) 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 \(\zeta(2m+1)\), a two-parameter generalization of the transformation formula for \(\log\eta(z)\), identities relating \(\zeta(2N+1),\zeta(4N+1),\dots,\zeta(2Nm+1)\) for odd \(N\), and transcendence or irrationality criteria of Zudilin- and Rivoal-type.

The explicitly proposed multivariate analogue is
\[
\sum_{n_{1},\ldots,n_{r}\ge 1}
\frac{P(n_{1},\ldots,n_{r})\,q^{\sum_{j=1}^{r} a_{j} n_{j}^{N_{j}}}}
{\prod_{j=1}^{r}\bigl(1-q^{n_{j}^{N_{j}}}\bigr)},
\]
where \(P\) is a polynomial, or a product of powers \(n_j^{M_j}\), and \(a_j\in(0,1]\), \(N_j\in\mathbb{N}\). 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 \((a_j,B_j)\) through
\[
\prod_{j} a_j B_j^{N_j}=\pi^{\sum (N_j+1)},
\]
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 [1801.09181].

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 \(q\)-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.

Source: https://www.emergentmind.com/topics/multivariate-generalized-lambert-series