Generating Function Fₖ,ₘ(q): Analysis & Applications

• Generating Function Fₖ,ₘ(q) is defined in multiple frameworks, including inverse orbit analysis of 3x+k maps where its rationality is linked to the Collatz conjecture.
• It serves in partition theory by enumerating two‐color partitions with provable positivity and connects to mock theta functions in limiting cases.
• The function also appears in binomial–hypergeometric sums, exhibiting C-finite structures and explicit linear recurrences that enable closed-form evaluations.

The notation $F_{k,m}(q)$ is used for concrete families of generating functions across several domains, including dynamical systems (inverse orbits of the $3x+k$ maps), partition theory (two-color partition generating functions), and binomial-hypergeometric structures. Its formal properties and analytic behavior are highly context dependent, reflecting deep connections to analytic number theory, combinatorics, and mathematical physics.

1. Definitions and Domain-Specific Forms

1. Inverse Orbit Generating Functions for $3x+k$ Maps:

Let $k\equiv\pm1 \pmod{6}$ and define the $3x+k$ map $T_k:\mathbb{Z}\to\mathbb{Z}$ by

$$T_k(n) = \begin{cases} \dfrac{3n+k}{2}, & n \text{ odd},\ \dfrac{n}{2}, & n \text{ even}. \end{cases}$$

The backward (inverse) $T_k$-orbit of $m\in\mathbb{N}^+$ is

$$\Omega_k^-(m) = \{ n>0 : T_k^{(\ell)}(n)=m\ \text{for some }\ell\ge0 \}.$$

The corresponding generating function, denoted both $F_{k,m}(q)$ and $f_{k,m}(q)$, is

$F_{k,m}(q) = \sum_{n\in\Omega_k^-(m)} q^n.$

This includes the fundamental $3x+1$ ("Collatz") case $k=1$ and $3x-1$ case $k=-1$ (Bell et al., 2014).

2. Two-Color Partition Generating Functions (Andrews–Bachraoui):

$$F_{k,m}(q) := \sum_{n\ge0} \frac{(q^{2n+2},\,q^{2n+2k};q^2)_\infty}{(q^{2n+1};q^2)_\infty^2}\;q^{m(2n+1)}.$$

Here, $(a;q)_n = \prod_{j=0}^{n-1}(1-aq^j)$ is the standard $q$-Pochhammer symbol. For $m=1$, $F_{k,1}(q)$ enumerates two-color partitions of integers with detailed part and multiplicity restrictions (Banerjee et al., 7 Jan 2026).

3. Alternating Binomial–Hypergeometric Sums:

$$F_{k,m}(q) := \sum_{j=0}^m \sum_{i=0}^j (-1)^{j-i} \binom{m}{j} \binom{j}{i} \binom{j+k+q i}{j+k}$$

with closed-form, single-sum, and hypergeometric series representations, as well as rational generating functions in $m$ (Mathar, 2023).

2. Analytic and Algebraic Properties

Inverse Orbit Functions for $3x+k$

• Holomorphy: $F_{k,m}(q)$ converges for $|q|<1$, with all singularities on the unit circle and none in $|q|<1$ (Bell et al., 2014).
• Natural Boundary Phenomenon: Applying the Pólya–Carlson dichotomy, $F_{k,m}$ is either rational or possesses the unit circle $|q|=1$ as a natural boundary.
• Explicit Orbit Sums and Rational Cases: For $k=1$, the Collatz scenario, $f_{1,m}(q)$ is rational precisely for $m\in\{1,2,4,8\}$ under the $3x+1$ conjecture, and otherwise has the unit circle as its natural boundary. Rationality in these exceptional cases is equivalent to the truth of the Collatz conjecture.
• Residue Class Structure: Rationality of $g(z)=\sum_{n\in S}z^n$ (union of finitely many backward orbits) occurs if and only if some set $X\subset\{0,1,\dots,|k|-1\}$, closed under multiplication by $2$ and $3$ modulo $|k|$, and $k_0$ exists so that $g(z)$ agrees with $\sum_{a\in X} z^a/(1-z^{|k|})$ for $n\ge k_0$ (Bell et al., 2014).

Two-Color Partition Functions

• Positivity: For $m=1$, Andrews–Bachraoui conjectured and verified for $1\leq k\leq10$ that all coefficients of $F_{k,1}(q)$ are non-negative.
• Product-to-Sum Expansions: $F_{k,1}(q)$ admits finite, rational product–to–sum expansions and partial fraction decompositions facilitating positivity proofs up to $k=10$.
• $q$-Binomial Quotient Form: The coefficients $G_{k,n}(q)$ in the expansion $F_{k,1}(q) = \sum_{n\ge0} G_{k,n}(q)\,q^{n+1}$ have the structure:

$$G_{k,n}(q) = \frac{1}{1-q^{2n+1}} \frac{\binom{2k+2n-2}{2n}_q}{\binom{k+n-1}{n}_{q^2}}.$$

• Limiting Behavior: As $k\to\infty$, $F_{k,1}(q)\to q \omega(q)$, where $\omega(q)$ is Ramanujan's third order mock theta function. The precise error is $q^{2k+1} E_k(q)$ for some explicit $E_k(q)\in\mathbb{Z}[[q]]$, matching coefficients up to $q^{2k}$ (Banerjee et al., 7 Jan 2026).

Binomial–Hypergeometric Generating Functions

• C-finite Structure: For each $k$, $F_{k,m}(q)$ is C-finite in $m$, admitting rational generating functions in terms of $t$:

$$G_k(t;q) = \frac{(1-t)^k Q_{k,q}(t)}{(1-(q+1)t)^{k+1}}$$

for some explicit polynomial $Q_{k,q}(t)$.

• Linear Recurrence: $F_{k,m}(q)$ satisfies an explicit order-$(k+2)$ linear recurrence in $m$ with constant coefficients.
• Hypergeometric Evaluation: $F_{k,m}(q)$ can be written as a terminating hypergeometric series of the Saalschützian type, making closed-form evaluation in small cases practical (Mathar, 2023).

3. Explicit Examples and Special Cases

$k$, $m$	$F_{k,m}(q)$ structure	Notable cases/limits
$k=1$, $m\leq8$ (Collatz)	Rational in $q$ if and only if $m\in\{1,2,4,8\}$	Equates to $3x+1$ conjecture (Bell et al., 2014)
Two-color, $m=1$, $k\to\infty$	$q \omega(q)$ (mock theta function)	$F_{k,1}(q)$ matches $q \omega(q)$ through $q^{2k}$ (Banerjee et al., 7 Jan 2026)
Binomial–Hypergeometric	Saalschützian ${}_{q+2}F_{q+1}(1)$	Admits explicit rational generating functions (Mathar, 2023)

Orbit Generating Functions, Small $m$: For $k=1$: $$\begin{aligned} f_{1,1}(q) &= \frac{q}{1-q} \ f_{1,2}(q) &= \frac{q^2}{1 - q^2} \ f_{1,4}(q) &= \frac{q^4}{1 - q^4} - q - q^2 \ f_{1,8}(q) &= \frac{q^8}{1 - q^8} - q^2 - q^4 \end{aligned}$$ each manifestly rational in $q$.

Two-Color Partition, $k=2$, $m=4$: The only $m\ne1$ case for which positivity is verified, with a specialized $q$-series proof (Banerjee et al., 7 Jan 2026).

Small-$k$ Binomial–Hypergeometric Examples: For $k=0$, $G_0(t;q)=1/(1-(q+1)t)$; for $k=1$, $G_1(t;q)=(1+(q-1)t)/(1-(q+1)t)^2$.

4. Connections to Broader Mathematical and Physical Contexts

Dynamical Systems:

The $F_{k,m}(q)$ functions associated to $3x+k$ maps encode inverse orbits, making them central in analytic formulations of orbit structure and conjectures (e.g., Collatz). Their analytic continuation and rationality dichotomies are linked directly to deep unsolved problems and exploit the Skolem–Mahler–Lech and Pólya–Carlson theorems (Bell et al., 2014).

Partition Theory and Mock Modular Forms:

The Andrews–Bachraoui $F_{k,m}(q)$ series relate partition enumeration to positivity phenomena, and in the $k\to\infty$ limit, they reproduce mock theta function expansions, indicating a bridge to mock modular forms and the theory of $q$-hypergeometric identities (Banerjee et al., 7 Jan 2026).

Binomial Identities and Hypergeometric Sums:

Explicit C-finite and recurrence structures connect these generating functions to the algebraic theory of linear recurrences and to algorithmic computation of sums involving binomial and generalized hypergeometric coefficients (Mathar, 2023).

5. Structural Theorems and Recurrence Properties

Pólya–Carlson Dichotomy:

For integer-coefficient series with radius of convergence $1$, the unit circle is either a natural boundary or the series is rational. In the $3x+k$ context, this dichotomy enables equivalences between analytic properties and orbit periodicity/reducibility (Bell et al., 2014).

Skolem–Mahler–Lech Theorem:

Rationality of orbit GFs is equivalent to ultimate periodicity—here realized as residue class unions stable under monoid actions (multiplication by $2$ and $3$ modulo $|k|$).

Explicit Recurrence Relations for Binomial-Hypergeometric GF:

For the $a_{k,q}(m)$ sequences of (Mathar, 2023), the order-$(k+2)$ linear recurrence

$(1-(q+1)E)^{k+1}(1-E)F_{k,m}(q)=0$

governs all such $F_{k,m}(q)$ for fixed $k$, $q$.

Algorithmic Positivity:

For the Andrews–Bachraoui functions, algorithmic and partial fraction methods allow finite computation verifying nonnegativity of coefficients up to large $k$ (Banerjee et al., 7 Jan 2026).

6. Open Problems and Contemporary Research Directions

• Collatz Rationality Equivalence: Rationality in the four exceptional Collatz backward-orbit generating functions remains equivalent to the $3x+1$ conjecture, maintaining the status of analytic approaches tightly connected to one of mathematics' enduring open problems.
• Positivity in Two-Color Partition Series: Positivity is established for $F_{k,1}(q)$ to $k=10$, but the general, all-$k$, or arbitrary $m$ cases are open (Banerjee et al., 7 Jan 2026).
• Binomial–Binomial Quotient Positivity: The nonnegativity of the quotient

$$\frac{\binom{2k+2n-2}{2n}_q}{\binom{k+n-1}{n}_{q^2}}$$

after division by $1-q^{2n+1}$ remains unproven in general, and a direct combinatorial or analytic proof is lacking.

7. Summary Table of Key Structures

Context	Definition of $F_{k,m}(q)$	Main property highlights
$3x+k$ inverse orbits	$\sum_{n\in\Omega_k^-(m)} q^n$	Holomorphic in $|q|<1$; rationality$\Leftrightarrow$periodicity, Collatz conjecture equivalence (Bell et al., 2014)
Two-color partitions	$$\sum_{n\ge0} \frac{(q^{2n+2},q^{2n+2k};q^2)_\infty}{(q^{2n+1};q^2)_\infty^2} q^{m(2n+1)}$$	Coefficient positivity for $m=1$, limiting mock theta ($k\to\infty$) (Banerjee et al., 7 Jan 2026)
Binomial–hypergeometric sums	$$\sum_{j=0}^m\sum_{i=0}^j (-1)^{j-i}\binom{m}{j}\binom{j}{i}\binom{j+k+qi}{j+k}$$	C-finite, rational ordinary generating functions, explicit recurrences (Mathar, 2023)

Each occurrence of $F_{k,m}(q)$ demonstrates the rich interplay between generating function theory, recurrence structure, analytic continuation, and combinatorial enumeration. These functions epitomize the diversity and depth of generating function methods in contemporary mathematics.