Generalized Lambert Series

• Generalized Lambert series are extensions of classical Lambert series that encode divisor-sum transforms and support modular-type, arithmetic, and multivariate generalizations.
• They include shifted, analytic two-parameter, and character-twisted forms with practical applications in partition theory, asymptotic analysis, and computation acceleration.
• Their factorization and transformation properties lead to new representations of quasi-modular forms, Ramanujan-type identities, and operator-based identities in number theory.

Generalized Lambert series are extensions of the classical Lambert series

$$L_f(q):=\sum_{n\ge 1}\frac{f(n)q^n}{1-q^n}=\sum_{m\ge 1}(f*1)(m)q^m,$$

viewed either as formal $q$-series encoding divisor-sum transforms, or as analytic objects admitting modular-type transformations, asymptotic expansions, factorization theorems, and multivariable or twisted variants (Schmidt, 2020). In current usage, the term covers several related constructions: shifts to arithmetic progressions, weighted divisor sums, higher-power and multi-parameter exponential deformations, double and multivariate sums, and character twists. A recent example shows that a seemingly complicated double Lambert series is exactly another presentation of the generating function for $\sigma_1(N)$, and therefore yields a new representation of the quasi-modular form $E_2(q)$ (Kumar et al., 20 May 2026).

1. Classical prototype and principal families

The basic arithmetic mechanism is the divisor-sum identity

$$L_f(q)=\sum_{n\ge1}\frac{f(n)q^n}{1-q^n}=\sum_{m\ge1}(f*1)(m)q^m,$$

so the coefficient of $q^m$ is $\sum_{d\mid m}f(d)$ (Schmidt, 2020). A standard generalization shifts the support to an arithmetic progression,

$$L_f(\alpha,\beta;q):=\sum_{n\ge1}\frac{f(n)\,q^{\alpha n-\beta}}{1-q^{\alpha n-\beta}},\qquad 0\le \beta<\alpha,$$

with coefficients

$$[q^n]\,L_f(\alpha,\beta;q)=\sum_{\alpha d-\beta\mid n}f(d)$$

(Schmidt, 2020). Another pervasive analytic form is

$$\mathscr L_q(s,x)=\sum_{k=1}^\infty \frac{k^s q^{kx}}{1-q^k},$$

which interpolates divisor-generating functions and polylogarithmic asymptotics near $q$0 (Banerjee et al., 2016). A third large family is built from generalized divisor functions, for example

$q$1

and its classical $q$2 specialization $q$3 (Banerjee et al., 2023). More recently, the literature also studies character-twisted series such as

$q$4

whose coefficients are twisted divisor sums and whose $q$5 behavior is resurgent (Broadhurst et al., 28 Jul 2025).

Family	Representative form	Arithmetic or analytic role
Classical Lambert series	$q$6	Dirichlet convolution with $q$7
Shifted generalized series	$q$8	Divisors in an arithmetic progression
Analytic two-parameter series	$q$9	Divisor generating functions and $\sigma_1(N)$0 asymptotics
Generalized divisor exponential series	$\sigma_1(N)$1	Voronoï-type transformations and modular-type identities
Character-twisted series	$\sigma_1(N)$2	Twisted Eisenstein and resurgent structures

This diversity is not merely terminological. It reflects three persistent viewpoints: formal divisor-sum generation, modular or quasi-modular transformation theory, and operator-theoretic or multivariable deformation. The subject therefore sits simultaneously in multiplicative number theory, $\sigma_1(N)$3-series, and automorphic analysis.

2. Factorization, restricted divisors, and algebraic structure

A major structural development is the factorization of Lambert series against a fixed power series $\sigma_1(N)$4. In the classical setting,

$\sigma_1(N)$5

and the universal coefficient formula is

$\sigma_1(N)$6

(Merca et al., 2017). This makes the factorization pair $\sigma_1(N)$7 a lower-triangular operator translating divisor sums into partition-theoretic or convolutional data.

For shifted generalized Lambert series, the factorization persists with modified kernels. One explicit formula is

$\sigma_1(N)$8

for

$\sigma_1(N)$9

which extends the ordinary theory to exponents supported on arithmetic progressions (Merca et al., 2017). In this framework, coefficients become restricted divisor sums such as

$E_2(q)$0

rather than the unrestricted $E_2(q)$1 (Merca et al., 2017).

Derivative methods yield a second algebraic layer. For

$E_2(q)$2

higher $E_2(q)$3-derivatives can be expanded through Stirling-number formulas and re-expressed in terms of bounded-index divisor sums

$E_2(q)$4

(Schmidt, 2017). The resulting identities express $E_2(q)$5 as finite linear combinations of polynomially weighted bounded-index divisor sums. This suggests that generalized Lambert series are not only divisor-sum generating functions; they also support a derivative-generated hierarchy of finite arithmetic expansions.

The significance of these factorizations is twofold. On the additive side, choices such as $E_2(q)$6 connect generalized Lambert series to partition statistics. On the multiplicative side, the same formalism accommodates $E_2(q)$7, $E_2(q)$8, $E_2(q)$9, $$L_f(q)=\sum_{n\ge1}\frac{f(n)q^n}{1-q^n}=\sum_{m\ge1}(f*1)(m)q^m,$$0, $$L_f(q)=\sum_{n\ge1}\frac{f(n)q^n}{1-q^n}=\sum_{m\ge1}(f*1)(m)q^m,$$1, $$L_f(q)=\sum_{n\ge1}\frac{f(n)q^n}{1-q^n}=\sum_{m\ge1}(f*1)(m)q^m,$$2, and Dirichlet convolutions of such functions (Merca et al., 2017, Merca et al., 2017). This duality between partitions and multiplicative functions is one of the defining structural features of the subject.

3. Transformation theory, modularity, and generalized divisor kernels

A central analytic branch studies generalized Lambert series through Mellin inversion, contour shifting, Hurwitz zeta functional equations, Voronoï summation, and specialized kernels. One important family is

$$L_f(q)=\sum_{n\ge1}\frac{f(n)q^n}{1-q^n}=\sum_{m\ge1}(f*1)(m)q^m,$$3

whose transformation was extended to arbitrary $$L_f(q)=\sum_{n\ge1}\frac{f(n)q^n}{1-q^n}=\sum_{m\ge1}(f*1)(m)q^m,$$4 and $$L_f(q)=\sum_{n\ge1}\frac{f(n)q^n}{1-q^n}=\sum_{m\ge1}(f*1)(m)q^m,$$5, and then used to obtain new Ramanujan-type identities relating $$L_f(q)=\sum_{n\ge1}\frac{f(n)q^n}{1-q^n}=\sum_{m\ge1}(f*1)(m)q^m,$$6 and $$L_f(q)=\sum_{n\ge1}\frac{f(n)q^n}{1-q^n}=\sum_{m\ge1}(f*1)(m)q^m,$$7 for odd $$L_f(q)=\sum_{n\ge1}\frac{f(n)q^n}{1-q^n}=\sum_{m\ge1}(f*1)(m)q^m,$$8, together with complementary even-$$L_f(q)=\sum_{n\ge1}\frac{f(n)q^n}{1-q^n}=\sum_{m\ge1}(f*1)(m)q^m,$$9 formulas generalizing Wigert’s transformation (Dixit et al., 2017). A related two-parameter deformation,

$q^m$0

was analyzed by combining Mellin inversion, contour shifting, Hurwitz zeta functional equations, and Raabe’s integral, producing two-parameter generalizations of Ramanujan’s formula and the transformation law for $q^m$1 (Dixit et al., 2018).

For the classical divisor family,

$q^m$2

an exact transformation called the master identity was derived for $q^m$3 and $q^m$4 (Dixit et al., 2020). In this formulation, modularity is recovered for odd integral $q^m$5, while for even integral $q^m$6 the transformation acquires explicit hyperbolic-integral correction terms involving $q^m$7 and $q^m$8. The paper identifies this correction as the “precise obstruction to modularity” (Dixit et al., 2020). The same framework recovers Ramanujan’s odd-zeta formula, Dedekind eta transformations, and the Wigert–Bellman identity, and it extends further through a generalized Watson kernel and a two-variable generalization of the modified Bessel function (Dixit et al., 2020).

The generalized divisor function

$q^m$9

admits an $\sum_{d\mid m}f(d)$0-fold extension of this transformation theory (Banerjee et al., 2023). Starting from a Voronoï summation formula for $\sum_{d\mid m}f(d)$1, explicit transformations were obtained for

$\sum_{d\mid m}f(d)$2

initially for $\sum_{d\mid m}f(d)$3 and then, via analytic continuation and asymptotics of a generalized Meijer-$\sum_{d\mid m}f(d)$4 kernel, on larger $\sum_{d\mid m}f(d)$5-domains (Banerjee et al., 2023). Specializations produce a generalized Ramanujan formula for $\sum_{d\mid m}f(d)$6, extended higher Herglotz identities, generalized Dedekind eta transformations, Wigert’s transformation, and asymptotics for generalized power partitions (Banerjee et al., 2023). The $\sum_{d\mid m}f(d)$7 case is explicitly noted to play an important role in string theory scattering amplitudes (Banerjee et al., 2023).

The transformation theory therefore shows generalized Lambert series functioning as a common analytic language for divisor sums, Eisenstein phenomena, zeta-value identities, eta-transformations, and modular obstructions.

4. Partition theory, overpartitions, and rank-difference identities

Generalized Lambert series also act as generating devices for partition-theoretic statistics. In overpartition theory, for integers $\sum_{d\mid m}f(d)$8 and $\sum_{d\mid m}f(d)$9 with $$L_f(\alpha,\beta;q):=\sum_{n\ge1}\frac{f(n)\,q^{\alpha n-\beta}}{1-q^{\alpha n-\beta}},\qquad 0\le \beta<\alpha,$$0, one studies

$$L_f(\alpha,\beta;q):=\sum_{n\ge1}\frac{f(n)\,q^{\alpha n-\beta}}{1-q^{\alpha n-\beta}},\qquad 0\le \beta<\alpha,$$1

where

$$L_f(\alpha,\beta;q):=\sum_{n\ge1}\frac{f(n)\,q^{\alpha n-\beta}}{1-q^{\alpha n-\beta}},\qquad 0\le \beta<\alpha,$$2

packages weighted divisor sums over divisors in a fixed arithmetic progression (Merca, 2021). The corresponding overpartition statistic is

$$L_f(\alpha,\beta;q):=\sum_{n\ge1}\frac{f(n)\,q^{\alpha n-\beta}}{1-q^{\alpha n-\beta}},\qquad 0\le \beta<\alpha,$$3

where $$L_f(\alpha,\beta;q):=\sum_{n\ge1}\frac{f(n)\,q^{\alpha n-\beta}}{1-q^{\alpha n-\beta}},\qquad 0\le \beta<\alpha,$$4 is the total number of non-overlined parts equal to $$L_f(\alpha,\beta;q):=\sum_{n\ge1}\frac{f(n)\,q^{\alpha n-\beta}}{1-q^{\alpha n-\beta}},\qquad 0\le \beta<\alpha,$$5 among all overpartitions of $$L_f(\alpha,\beta;q):=\sum_{n\ge1}\frac{f(n)\,q^{\alpha n-\beta}}{1-q^{\alpha n-\beta}},\qquad 0\le \beta<\alpha,$$6 (Merca, 2021). Its generating function is

$$L_f(\alpha,\beta;q):=\sum_{n\ge1}\frac{f(n)\,q^{\alpha n-\beta}}{1-q^{\alpha n-\beta}},\qquad 0\le \beta<\alpha,$$7

(Merca, 2021). This construction yields recurrences, inequalities, and explicit links between overpartition statistics and multiplicative functions such as $$L_f(\alpha,\beta;q):=\sum_{n\ge1}\frac{f(n)\,q^{\alpha n-\beta}}{1-q^{\alpha n-\beta}},\qquad 0\le \beta<\alpha,$$8, $$L_f(\alpha,\beta;q):=\sum_{n\ge1}\frac{f(n)\,q^{\alpha n-\beta}}{1-q^{\alpha n-\beta}},\qquad 0\le \beta<\alpha,$$9, $$[q^n]\,L_f(\alpha,\beta;q)=\sum_{\alpha d-\beta\mid n}f(d)$$0, $$[q^n]\,L_f(\alpha,\beta;q)=\sum_{\alpha d-\beta\mid n}f(d)$$1, and $$[q^n]\,L_f(\alpha,\beta;q)=\sum_{\alpha d-\beta\mid n}f(d)$$2 (Merca, 2021).

A second strand connects generalized Lambert series identities to Appell functions and rank-crank PDEs. A multivariable identity due to Chan, Watson, and Jackson type generalizations was used to derive Rank-Crank type PDEs for higher order Appell functions, extending the Atkin–Swinnerton-Dyer elliptic-function method (Chan et al., 2012). In this setting, generalized Lambert series identities are the $$[q^n]\,L_f(\alpha,\beta;q)=\sum_{\alpha d-\beta\mid n}f(d)$$3-series source of higher odd-order Rank-Crank PDEs and of explicit decompositions of Appell functions into Lambert-type pieces (Chan et al., 2012).

The same-pole and double-pole problem appears in overpartition rank differences. The $$[q^n]\,L_f(\alpha,\beta;q)=\sum_{\alpha d-\beta\mid n}f(d)$$4-series

$$[q^n]\,L_f(\alpha,\beta;q)=\sum_{\alpha d-\beta\mid n}f(d)$$5

was introduced precisely to organize generalized Lambert series with repeated or related poles (Wei et al., 2018). This produces new identities with single and double poles, is effective when $$[q^n]\,L_f(\alpha,\beta;q)=\sum_{\alpha d-\beta\mid n}f(d)$$6 is a unit root, and leads to 3-dissections of overpartition ranks modulo $$[q^n]\,L_f(\alpha,\beta;q)=\sum_{\alpha d-\beta\mid n}f(d)$$7 as well as identities involving the third order mock theta functions $$[q^n]\,L_f(\alpha,\beta;q)=\sum_{\alpha d-\beta\mid n}f(d)$$8 and $$[q^n]\,L_f(\alpha,\beta;q)=\sum_{\alpha d-\beta\mid n}f(d)$$9 (Wei et al., 2018).

These results show that generalized Lambert series are not merely arithmetic generating functions. They also control partition statistics, overpartition recurrences, mock-theta relations, and PDEs attached to Appell–Lerch structures.

5. Double sums, multivariate operators, and exact reductions

Recent work has emphasized genuinely higher-dimensional generalizations. A particularly sharp result settles a conjecture of Amdeberhan, Andrews, and Ballantine by proving

$$\mathscr L_q(s,x)=\sum_{k=1}^\infty \frac{k^s q^{kx}}{1-q^k},$$0

for positive integers $$\mathscr L_q(s,x)=\sum_{k=1}^\infty \frac{k^s q^{kx}}{1-q^k},$$1 (Kumar et al., 20 May 2026). The key base identity is

$$\mathscr L_q(s,x)=\sum_{k=1}^\infty \frac{k^s q^{kx}}{1-q^k},$$2

proved by reducing the double Lambert series to a single Lambert series via an elementary decomposition and a telescoping identity (Kumar et al., 20 May 2026). The same identity yields the representation

$$\mathscr L_q(s,x)=\sum_{k=1}^\infty \frac{k^s q^{kx}}{1-q^k},$$3

presented there as a new Lambert-series expression for the quasi-modular form $$\mathscr L_q(s,x)=\sum_{k=1}^\infty \frac{k^s q^{kx}}{1-q^k},$$4 (Kumar et al., 20 May 2026).

A different multivariate direction is operator-theoretic. Using the $$\mathscr L_q(s,x)=\sum_{k=1}^\infty \frac{k^s q^{kx}}{1-q^k},$$5-derivative $$\mathscr L_q(s,x)=\sum_{k=1}^\infty \frac{k^s q^{kx}}{1-q^k},$$6 and the Partial Theta operator

$$\mathscr L_q(s,x)=\sum_{k=1}^\infty \frac{k^s q^{kx}}{1-q^k},$$7

one obtains

$$\mathscr L_q(s,x)=\sum_{k=1}^\infty \frac{k^s q^{kx}}{1-q^k},$$8

and hence the generalized Lambert series

$$\mathscr L_q(s,x)=\sum_{k=1}^\infty \frac{k^s q^{kx}}{1-q^k},$$9

(López, 20 Jul 2025). The same framework defines generalized Lambert-Mehler, Lambert-Rogers, double-sum bivariate, and multivariate generalized Lambert series (López, 20 Jul 2025).

These constructions enlarge the scope of the subject beyond the divisor-sum prototype. A plausible implication is that “generalized Lambert series” now denotes not one canonical object but a stable pattern: a rational or $q$00-hypergeometric kernel that linearizes under a transformation, factorization, or operator identity.

6. Computation, asymptotics, resurgence, and terminological boundaries

From the computational viewpoint, acceleration formulas are a central theme. For

$q$01

a Rogers–Fine specialization yields a transformed series with $q$02 decay, described as Theta convergence (Arndt, 2012). The symmetry

$q$03

is immediate in this representation (Arndt, 2012). Specialization to $q$04, $q$05 recovers Clausen’s identity

$q$06

which illustrates the acceleration from geometric to $q$07-scale decay (Arndt, 2012).

Near $q$08, the generalized Lambert series

$q$09

admits a complete asymptotic expansion (Banerjee et al., 2016). At $q$10, the leading term recovers the classical Knopp-type asymptotic

$q$11

for $q$12 (Banerjee et al., 2016). The same expansion controls the asymptotics of the $q$13-gamma and $q$14-polygamma functions, the $q$15-Pochhammer symbol, and Jacobi theta functions (Banerjee et al., 2016).

The resurgent perspective extends this boundary analysis. For Lambert series twisted by Dirichlet characters, exact resurgent transseries were computed near $q$16, with perturbative and non-perturbative sectors related by Fricke inversion (Broadhurst et al., 28 Jul 2025). In special integer and parity-matched cases, the perturbative expansion terminates and the Lambert series becomes an iterated integral of twisted Eisenstein series; for generic parameters, the same formulas are interpreted as a quantum-modular version of Fricke involution (Broadhurst et al., 28 Jul 2025).

A persistent source of confusion is terminological. In several papers, “generalized Lambert” refers not to $q$17-series but to generalizations of the Lambert $q$18 function, such as inverses of $q$19, $q$20, or $q$21 (Mugnaini, 2014, Mező et al., 2014, Kreinin et al., 8 Apr 2025). Those works belong to the theory of generalized Lambert functions rather than generalized Lambert series. The distinction is substantive: the series literature concerns divisor sums, $q$22-products, partition statistics, and modular or resurgent transformations, whereas the function literature concerns inverse transcendental maps, branch structure, and Lagrange inversion.

In that precise sense, generalized Lambert series form a broad but coherent domain. Their unifying role is to encode arithmetic and combinatorial data in kernels that remain tractable under factorization, asymptotic analysis, modular transformation, and multivariate deformation.