Truncated Identity of Gauss

• The truncated identity of Gauss is a family of finite analogues that replace infinite theta-series with finite sums accompanied by explicit tail corrections.
• These identities lead to sign-alternating partition inequalities and provide combinatorial interpretations linking overpartitions with minimal excludant statistics.
• The truncated forms extend classical results, offering finite approximations of Gauss’s square-exponent theorem and facilitating studies in modular diagrams and coefficient positivity.

Searching arXiv for recent and foundational papers on the truncated identity of Gauss. The truncated identity of Gauss denotes a family of finite or partial-sum analogues of Gauss’s classical theta-series identities in $q$-series. In the literature surveyed here, the central objects are truncations of the product–sum formulas

$$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$

and

$$1+\sum_{j=0}^\infty(-1)^j q^{j(2j+1)}(1-q^{2j+1}) =\frac{(-q;q^2)_\infty\,(q^2;q^2)_\infty}{1},$$

together with related finitizations of Gauss’s square-exponent theorem and companion identities (Guo et al., 2012, Chern, 2018). These truncations replace an infinite theta series by a finite sum indexed by a parameter such as $k$, $L$, or $v$, and compensate for the truncation by an explicit tail series or by a finite identity equal to $1$ (Guo et al., 2012, Gireesh et al., 2022, Chern, 2018). Subsequent work has connected these formulas to overpartitions, partitions with distinct odd parts, modular Young diagrams, Durfee rectangles, coefficient-positivity problems, and combinatorial statistics such as the minimal excludant of non-overlined parts of an overpartition (He et al., 1 Sep 2025, Gireesh et al., 2022, Liu, 2024).

1. Classical Gauss identities and the meaning of truncation

Gauss’s classical theta-product identity appears in the form

$$1 \;+\; 2\sum_{j=1}^\infty(-1)^j q^{j^2} \;=\;\frac{(q;q)_\infty}{(-q;q)_\infty},$$

or equivalently

$$\frac{(-q;q)_\infty}{(q;q)_\infty} \Bigl(1 + 2\sum_{j=1}^\infty(-1)^j q^{j^2}\Bigr) \;=\;1$$

(Guo et al., 2012, He et al., 1 Sep 2025). A second Gauss identity is recorded as

$$1\;+\;\sum_{j=0}^\infty(-1)^j q^{j(2j+1)}\bigl(1-q^{2j+1}\bigr) \;=\;\frac{(-q;q^2)_\infty\,(q^2;q^2)_\infty}{1}$$

(Guo et al., 2012). Closely related formulations also appear as

$$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$0

in a partition-theoretic treatment (Gireesh et al., 2022), and as

$$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$1

for Gauss’s square-exponent theorem (Chern, 2018).

In this setting, truncation means cutting off the infinite theta series after finitely many terms. The resulting finite expression is not equal to the product side alone; rather, it is accompanied by an explicit remainder, tail, or correction term. The truncation parameter controls how many theta terms are retained, and letting that parameter tend to infinity recovers the original infinite identity (Guo et al., 2012, Gireesh et al., 2022, Chern, 2018).

2. Guo–Zeng truncations and explicit tail expansions

A foundational development is the pair of truncated identities established by Guo and Zeng. For any positive integer $$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$2, the triangular-number series admits the finite form

$$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$3

and the square-number series has the truncated analogue

$$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$4

(Guo et al., 2012).

A related formulation uses the parameter $$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$5: $$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$6 and

$$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$7

(Gireesh et al., 2022).

These identities are structurally important because they exhibit a finite theta sum on the left and a manifestly organized tail on the right. In the cited work, the tail is then reinterpreted either analytically, as a single $$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$8-series with explicit $$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$9-Pochhammer factors and $$1+\sum_{j=0}^\infty(-1)^j q^{j(2j+1)}(1-q^{2j+1}) =\frac{(-q;q^2)_\infty\,(q^2;q^2)_\infty}{1},$$0-binomial coefficients, or combinatorially, as the generating function of partitions subject to modular and Durfee-type constraints (Guo et al., 2012, Gireesh et al., 2022).

3. Partition inequalities and combinatorial consequences

The first major consequence of these truncations is a family of sign-alternating partition inequalities. Since the overpartition generating function satisfies

$$1+\sum_{j=0}^\infty(-1)^j q^{j(2j+1)}(1-q^{2j+1}) =\frac{(-q;q^2)_\infty\,(q^2;q^2)_\infty}{1},$$1

equating coefficients in the truncated triangular identity yields, for all $$1+\sum_{j=0}^\infty(-1)^j q^{j(2j+1)}(1-q^{2j+1}) =\frac{(-q;q^2)_\infty\,(q^2;q^2)_\infty}{1},$$2,

$$1+\sum_{j=0}^\infty(-1)^j q^{j(2j+1)}(1-q^{2j+1}) =\frac{(-q;q^2)_\infty\,(q^2;q^2)_\infty}{1},$$3

with strict inequality if $$1+\sum_{j=0}^\infty(-1)^j q^{j(2j+1)}(1-q^{2j+1}) =\frac{(-q;q^2)_\infty\,(q^2;q^2)_\infty}{1},$$4 (Guo et al., 2012). In particular,

$$1+\sum_{j=0}^\infty(-1)^j q^{j(2j+1)}(1-q^{2j+1}) =\frac{(-q;q^2)_\infty\,(q^2;q^2)_\infty}{1},$$5

(Guo et al., 2012).

Similarly, if $$1+\sum_{j=0}^\infty(-1)^j q^{j(2j+1)}(1-q^{2j+1}) =\frac{(-q;q^2)_\infty\,(q^2;q^2)_\infty}{1},$$6 denotes the number of partitions of $$1+\sum_{j=0}^\infty(-1)^j q^{j(2j+1)}(1-q^{2j+1}) =\frac{(-q;q^2)_\infty\,(q^2;q^2)_\infty}{1},$$7 in which odd parts are all distinct, with generating function

$$1+\sum_{j=0}^\infty(-1)^j q^{j(2j+1)}(1-q^{2j+1}) =\frac{(-q;q^2)_\infty\,(q^2;q^2)_\infty}{1},$$8

then the square-number truncation implies, for all $$1+\sum_{j=0}^\infty(-1)^j q^{j(2j+1)}(1-q^{2j+1}) =\frac{(-q;q^2)_\infty\,(q^2;q^2)_\infty}{1},$$9,

$k$0

with strict inequality when $k$1 (Guo et al., 2012).

These inequalities were generalized in several directions. One common generalization introduces $k$2 by

$k$3

together with a conjectured sign-alternating truncation inequality (Guo et al., 2012). Another conjectural extension concerns the three-flavor partition function $k$4 defined by

$k$5

for which a truncation of Jacobi’s recurrence is conjectured to have alternating sign (Guo et al., 2012).

A different partition-theoretic realization uses modular Young diagrams and $k$6-Durfee rectangles. For fixed integers $k$7 and $k$8, the quantity $k$9 counts partitions of $L$0 into parts $L$1 such that every residue-class part $L$2 occurs at least once, and in the $L$3-modular graph the parts below the $L$4-Durfee rectangle are all strictly less than its width (Gireesh et al., 2022). Its generating function is

$L$5

(Gireesh et al., 2022). By inserting this into the truncated Gauss identities, one obtains exact difference-of-partition-count formulas and inequalities such as

$L$6

under the stated parity conditions (Gireesh et al., 2022).

4. Minimal excludant interpretation for overpartitions

A more recent development gives a direct combinatorial interpretation of the truncated Gauss identity in terms of overpartitions and a minimal-excludant statistic (He et al., 1 Sep 2025). An overpartition of $L$7 is a partition of $L$8 in which the first occurrence of each part may be overlined, and the total order is

$L$9

(He et al., 1 Sep 2025).

For fixed $v$0 with $v$1, the statistic $v$2 is defined to be the smallest positive integer that is congruent to $v$3 and does not occur among the non-overlined parts of $v$4. The specialization used in the paper is $v$5, so $v$6 is the least odd integer missing as a non-overlined part (He et al., 1 Sep 2025).

For $v$7 and $v$8, let $v$9 denote the number of overpartitions $1$0 of $1$1 satisfying

$1$2

Then the main theorem states that for all integers $1$3 and $1$4,

$1$5

(He et al., 1 Sep 2025). Thus the truncated Gauss combination does not merely have nonnegative value; it counts twice a specific family of overpartitions.

The generating-function proof is based on two lemmas: $1$6 for each fixed $1$7 and $1$8, and

$1$9

for every $$1 \;+\; 2\sum_{j=1}^\infty(-1)^j q^{j^2} \;=\;\frac{(q;q)_\infty}{(-q;q)_\infty},$$0 (He et al., 1 Sep 2025). Substituting these into the truncated sum yields a telescoping expression whose remaining term is exactly $$1 \;+\; 2\sum_{j=1}^\infty(-1)^j q^{j^2} \;=\;\frac{(q;q)_\infty}{(-q;q)_\infty},$$1 (He et al., 1 Sep 2025).

A worked example is given for $$1 \;+\; 2\sum_{j=1}^\infty(-1)^j q^{j^2} \;=\;\frac{(q;q)_\infty}{(-q;q)_\infty},$$2, $$1 \;+\; 2\sum_{j=1}^\infty(-1)^j q^{j^2} \;=\;\frac{(q;q)_\infty}{(-q;q)_\infty},$$3. Since $$1 \;+\; 2\sum_{j=1}^\infty(-1)^j q^{j^2} \;=\;\frac{(q;q)_\infty}{(-q;q)_\infty},$$4 and $$1 \;+\; 2\sum_{j=1}^\infty(-1)^j q^{j^2} \;=\;\frac{(q;q)_\infty}{(-q;q)_\infty},$$5,

$$1 \;+\; 2\sum_{j=1}^\infty(-1)^j q^{j^2} \;=\;\frac{(q;q)_\infty}{(-q;q)_\infty},$$6

On the combinatorial side, only the overpartition $$1 \;+\; 2\sum_{j=1}^\infty(-1)^j q^{j^2} \;=\;\frac{(q;q)_\infty}{(-q;q)_\infty},$$7 has smallest missing non-overlined odd part equal to $$1 \;+\; 2\sum_{j=1}^\infty(-1)^j q^{j^2} \;=\;\frac{(q;q)_\infty}{(-q;q)_\infty},$$8, so $$1 \;+\; 2\sum_{j=1}^\infty(-1)^j q^{j^2} \;=\;\frac{(q;q)_\infty}{(-q;q)_\infty},$$9, and the right-hand side is again $$\frac{(-q;q)_\infty}{(q;q)_\infty} \Bigl(1 + 2\sum_{j=1}^\infty(-1)^j q^{j^2}\Bigr) \;=\;1$$0 (He et al., 1 Sep 2025).

5. Finitizations of Gauss’s square-exponent theorem

A parallel line of research studies finite identities equal to $$\frac{(-q;q)_\infty}{(q;q)_\infty} \Bigl(1 + 2\sum_{j=1}^\infty(-1)^j q^{j^2}\Bigr) \;=\;1$$1 that truncate Gauss’s square-exponent theorem. In its standard form,

$$\frac{(-q;q)_\infty}{(q;q)_\infty} \Bigl(1 + 2\sum_{j=1}^\infty(-1)^j q^{j^2}\Bigr) \;=\;1$$2

(Chern, 2018). J.-C. Liu obtained three finite truncations, each summing over $$\frac{(-q;q)_\infty}{(q;q)_\infty} \Bigl(1 + 2\sum_{j=1}^\infty(-1)^j q^{j^2}\Bigr) \;=\;1$$3 from $$\frac{(-q;q)_\infty}{(q;q)_\infty} \Bigl(1 + 2\sum_{j=1}^\infty(-1)^j q^{j^2}\Bigr) \;=\;1$$4 to $$\frac{(-q;q)_\infty}{(q;q)_\infty} \Bigl(1 + 2\sum_{j=1}^\infty(-1)^j q^{j^2}\Bigr) \;=\;1$$5 and each having right-hand side exactly $$\frac{(-q;q)_\infty}{(q;q)_\infty} \Bigl(1 + 2\sum_{j=1}^\infty(-1)^j q^{j^2}\Bigr) \;=\;1$$6: $$\frac{(-q;q)_\infty}{(q;q)_\infty} \Bigl(1 + 2\sum_{j=1}^\infty(-1)^j q^{j^2}\Bigr) \;=\;1$$7

$$\frac{(-q;q)_\infty}{(q;q)_\infty} \Bigl(1 + 2\sum_{j=1}^\infty(-1)^j q^{j^2}\Bigr) \;=\;1$$8

and

$$\frac{(-q;q)_\infty}{(q;q)_\infty} \Bigl(1 + 2\sum_{j=1}^\infty(-1)^j q^{j^2}\Bigr) \;=\;1$$9

(Chern, 2018).

Chern added two new truncations: $$1\;+\;\sum_{j=0}^\infty(-1)^j q^{j(2j+1)}\bigl(1-q^{2j+1}\bigr) \;=\;\frac{(-q;q^2)_\infty\,(q^2;q^2)_\infty}{1}$$0 valid for all integers $$1\;+\;\sum_{j=0}^\infty(-1)^j q^{j(2j+1)}\bigl(1-q^{2j+1}\bigr) \;=\;\frac{(-q;q^2)_\infty\,(q^2;q^2)_\infty}{1}$$1, and

$$1\;+\;\sum_{j=0}^\infty(-1)^j q^{j(2j+1)}\bigl(1-q^{2j+1}\bigr) \;=\;\frac{(-q;q^2)_\infty\,(q^2;q^2)_\infty}{1}$$2

(Chern, 2018).

These formulas are obtained from finite-form $$1\;+\;\sum_{j=0}^\infty(-1)^j q^{j(2j+1)}\bigl(1-q^{2j+1}\bigr) \;=\;\frac{(-q;q^2)_\infty\,(q^2;q^2)_\infty}{1}$$3-identities by the substitutions

$$1\;+\;\sum_{j=0}^\infty(-1)^j q^{j(2j+1)}\bigl(1-q^{2j+1}\bigr) \;=\;\frac{(-q;q^2)_\infty\,(q^2;q^2)_\infty}{1}$$4

combined with the transformation behavior of $$1\;+\;\sum_{j=0}^\infty(-1)^j q^{j(2j+1)}\bigl(1-q^{2j+1}\bigr) \;=\;\frac{(-q;q^2)_\infty\,(q^2;q^2)_\infty}{1}$$5 and $$1\;+\;\sum_{j=0}^\infty(-1)^j q^{j(2j+1)}\bigl(1-q^{2j+1}\bigr) \;=\;\frac{(-q;q^2)_\infty\,(q^2;q^2)_\infty}{1}$$6 under $$1\;+\;\sum_{j=0}^\infty(-1)^j q^{j(2j+1)}\bigl(1-q^{2j+1}\bigr) \;=\;\frac{(-q;q^2)_\infty\,(q^2;q^2)_\infty}{1}$$7 (Chern, 2018). Letting $$1\;+\;\sum_{j=0}^\infty(-1)^j q^{j(2j+1)}\bigl(1-q^{2j+1}\bigr) \;=\;\frac{(-q;q^2)_\infty\,(q^2;q^2)_\infty}{1}$$8, the finite sums extend to all integers and the extra $$1\;+\;\sum_{j=0}^\infty(-1)^j q^{j(2j+1)}\bigl(1-q^{2j+1}\bigr) \;=\;\frac{(-q;q^2)_\infty\,(q^2;q^2)_\infty}{1}$$9-binomial factors tend to unity, recovering Gauss’s full theorem (Chern, 2018).

Chern also introduced multiple-summation extensions $$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$00 and $$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$01, showed by creative telescoping that

$$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$02

and gave explicit formulas for $$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$03 and $$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$04, while noting that no closed form for general $$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$05 is known (Chern, 2018). The paper explicitly remarks that a direct bijective or combinatorial interpretation of identities such as $$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$06 or $$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$07 remains unknown (Chern, 2018).

6. Later refinements: new truncations and coefficient positivity

Further work produced additional truncated expansions for Gauss-type series. One paper gives three new expansions for partial sums of Gauss’s triangular series, motivated by Andrews–Merca and Guo–Zeng and derived using summation formulas from Zhi-Guo Liu (He, 2018). Among them are

$$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$08

$$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$09

and three equivalent expansions for

$$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$10

(He, 2018). From one of these, the paper deduces that for any nonnegative integer $$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$11 and any truncation order $$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$12,

$$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$13

(He, 2018).

Another refinement concerns coefficient positivity. Liu places truncated Gauss identities into a larger family of truncated product–sum identities and studies the coefficients after division by

$$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$14

(Liu, 2024). In the Gauss specialization

$$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$15

the truncated generating function is

$$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$16

Corollary 2.3 asserts that there exists an explicit integer $$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$17, with

$$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$18

such that

$$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$19

(Liu, 2024). The proof uses a partial-fraction or periodic-function decomposition based on a lemma of Pólya–Szegő and then a nine-case piecewise quadratic analysis (Liu, 2024). Small-$$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$20 calculations show that positivity does not hold initially; for instance, when $$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$21,

$$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$22

so the coefficients are not nonnegative (Liu, 2024).

This development clarifies a common misconception. The positivity phenomena associated with truncated Gauss identities are not unconditional for arbitrary truncation depth; in this coefficientwise setting they emerge only beyond an explicit threshold (Liu, 2024). By contrast, the partition inequalities derived from the Guo–Zeng-type truncations hold uniformly for all $$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$23 in the stated forms (Guo et al., 2012).

7. Related directions and scope of the term

The phrase “truncated identity of Gauss” has also appeared in adjacent contexts. One paper studies truncated Gauss sums at rational arguments for factorization, defining

$$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$24

and states a “Truncated identity” asserting that for sufficiently large truncation $$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$25,

$$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$26

(Wölk et al., 2012). This is a distinct use of the phrase, tied to constructive interference and factor detection rather than to theta-product truncations.

A plausible implication is that the modern $$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$27-series literature uses “truncated identity of Gauss” primarily for finitizations of Gauss’s classical theta identities, while other areas use similar language for truncated Gauss-type sums with different analytic and algorithmic purposes. Within the $$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$28-series setting, the subject now includes explicit truncation formulas, partition inequalities, modular-graph models, minimal-excludant statistics on overpartitions, finite analogues of Gauss’s square-exponent theorem, and coefficient-positivity refinements (Guo et al., 2012, Gireesh et al., 2022, He et al., 1 Sep 2025, Chern, 2018, Liu, 2024).

Several open directions remain explicit in the literature. For the square-exponent truncations, direct bijective or combinatorial proofs are still unknown in key cases, and no closed form for the multiple-summation families is known for general $$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$29 (Chern, 2018). For partition-theoretic truncations, conjectural inequalities for broader classes such as $$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$30 and $$1+2\sum_{j=1}^\infty(-1)^j q^{j^2}=\frac{(q;q)_\infty}{(-q;q)_\infty}$$31 remain part of the program initiated by the truncated Gauss identities (Guo et al., 2012).