Paired Functions: A Conceptual Overview

• Paired functions are defined as structured pairs linking two functions via algebraic, analytic, combinatorial, or statistical relations, with examples like sine-cosine and Legendre dual pairs.
• They employ methodologies ranging from satisfying sine-addition equations and duality transformations to constructing bijections on discrete sets using n-adic and characteristic-function techniques.
• Applications span functional data analysis, Boolean functions, and many-body physics, revealing their versatility in modeling periodic behavior, duality, and quantum entanglement.

“Paired functions” is not a single universal notion. In contemporary literature, the expression is used for several non-equivalent constructions: function pairs linked by addition laws or translations, functions related by the Legendre transformation, reproducing pairs of weakly measurable functions, Boolean “paired” functions on the slice, pairs of functions arising in functional data analysis, and pairing functions governing fermionic many-body states and their entanglement spectra (Himmel, 2021, Kolt et al., 2022, Antoine et al., 2015, Kiermaier et al., 3 Oct 2025, Meyer, 29 Sep 2025, Dubail et al., 2011). The common feature is the imposition of a structured relation between two functions or between a function and a canonical partner, but the underlying algebraic, analytic, combinatorial, statistical, and physical meanings differ substantially.

1. Pairable functions and Cauchy pairs

In the functional-equation literature, a function $f$ is called pairable with $g$ if there exists a period function $T:I^2\to\mathbb{R}$ such that $g(x)=f(x+T)$ and the pair $(f,g)$ satisfies the sine-addition-type equation

$f(x+y)=f(x)g(y)+f(y)g(x),$

for all $x,y\in I$, where $I\subset\mathbb{R}$ is an interval closed under addition (Himmel, 2021). The associated cosine-type relation is

$g(x+y)=g(x)g(y)-f(x)f(y).$

A pair $(f,g)$ is called a Cauchy pair with respect to a given Cauchy functional equation when $g$0, and in some cases $g$1, is a regular solution of an additive, multiplicative, exponential, or logarithmic Cauchy equation (Himmel, 2021).

This framework extends the classical trigonometric model. The paper explicitly identifies classical paired functions such as $g$2 as instances where the period is constant, while “pairability” permits a non-constant period function and therefore a broader class of solutions (Himmel, 2021). The sine and cosine representers are given by

$g$3

The representers are described as uniquely determined away from zeros (Himmel, 2021).

The explicit examples in this literature emphasize that the period function need not be constant. For the additive case $g$4,

$g$5

while for the exponential case $g$6, pairability is possible only with constant period

$g$7

The paper also states that Euler’s identity is interpreted as a property of certain pairable functions and finite cyclic groups, with the Gamma function and reflection formulas providing the relevant structural link (Himmel, 2021). This suggests a broad functional-equation viewpoint in which “paired” behavior is encoded by addition theorems and generalized periodicity rather than by a fixed named partner alone.

2. Legendre pairs and pair-sharing in complex analysis

A second major use of function pairing is Legendre duality. A Legendre-transformation pair consists of functions $g$8 and $g$9 linked by

$T:I^2\to\mathbb{R}$0

for convex $T:I^2\to\mathbb{R}$1, or, when $T:I^2\to\mathbb{R}$2 is invertible,

$T:I^2\to\mathbb{R}$3

The paper gives the equivalent parametric representation

$T:I^2\to\mathbb{R}$4

and describes the involution property

$T:I^2\to\mathbb{R}$5

as a general theorem (Kolt et al., 2022).

The tabulated examples include quadratic, power, exponential, logarithmic, trigonometric, and special-function pairs. Among the standard entries are

$T:I^2\to\mathbb{R}$6

$T:I^2\to\mathbb{R}$7

$T:I^2\to\mathbb{R}$8

and

$T:I^2\to\mathbb{R}$9

The same source provides a practical authentication criterion: for a candidate pair, the quantity

$g(x)=f(x+T)$0

should vanish identically on the dual domain (Kolt et al., 2022).

A different complex-analytic notion of pairing appears in Nevanlinna theory. Two meromorphic functions $g(x)=f(x+T)$1 and $g(x)=f(x+T)$2 are said to share a pair $g(x)=f(x+T)$3 if $g(x)=f(x+T)$4 and $g(x)=f(x+T)$5 have the same zeros, with the usual modifications when one of the values is $g(x)=f(x+T)$6 (Steinmetz, 2014). The paper classifies pairs $g(x)=f(x+T)$7 that share four pairs $g(x)=f(x+T)$8, $g(x)=f(x+T)$9, and a fifth pair $(f,g)$0 counting multiplicities, under the additional condition

$(f,g)$1

Under these hypotheses, either $(f,g)$2 and $(f,g)$3 are Möbius transformations of each other, or there exist Möbius transformations $(f,g)$4 and $(f,g)$5 and a non-constant entire function $(f,g)$6 such that

$(f,g)$7

where $(f,g)$8 are Gundersen’s prototype functions (Steinmetz, 2014). The same work connects the classification to the algebraic curve

$(f,g)$9

and records as open whether the additional analytic condition can be dropped (Steinmetz, 2014). Here the paired structure is not duality but shared value-pair geometry.

3. Pairing bijections on discrete sets

In combinatorics and theoretical computer science, a pairing function is a bijection $f(x+y)=f(x)g(y)+f(y)g(x),$0. One paper describes two general mechanisms for producing infinite families of such bijections: one based on $f(x+y)=f(x)g(y)+f(y)g(x),$1-adic valuations, and one based on characteristic functions of subsets of $f(x+y)=f(x)g(y)+f(y)g(x),$2 (Tarau, 2013). The $f(x+y)=f(x)g(y)+f(y)g(x),$3-adic mechanism yields a countable family of distinct pairing bijections, while the characteristic-function mechanism yields $f(x+y)=f(x)g(y)+f(y)g(x),$4 distinct pairing bijections (Tarau, 2013). The paper also notes that these constructions can be combined to generate families of permutations of $f(x+y)=f(x)g(y)+f(y)g(x),$5, and gives the classical $f(x+y)=f(x)g(y)+f(y)g(x),$6-adic example

$f(x+y)=f(x)g(y)+f(y)g(x),$7

A distinct line of work constructs a symmetric invertible binary pairing function on positive integers,

$f(x+y)=f(x)g(y)+f(y)g(x),$8

with the property $f(x+y)=f(x)g(y)+f(y)g(x),$9 (Xie, 2021). The symmetry follows from dependence on $x,y\in I$0 and $x,y\in I$1, and the paper gives a complete proof of bijectivity together with an explicit inverse formula (Xie, 2021). The stated motivation is the encoding of unordered pairs, in contrast with Cantor’s pairing function, which is not symmetric (Xie, 2021).

Binary proportional pairing functions generalize binary perfect pairing functions to settings in which the two inputs have lengths differing by a fixed proportion (Szudzik, 2018). The base-$x,y\in I$2 length is defined by

$x,y\in I$3

The paper presents a general construction from any non-decreasing unbounded function $x,y\in I$4, with pairing map

$x,y\in I$5

where $x,y\in I$6 is the smallest $x,y\in I$7 such that $x,y\in I$8 (Szudzik, 2018). Specializing to

$x,y\in I$9

produces the proportional pairing function

$I\subset\mathbb{R}$0

The paper identifies binary perfect pairing as the special case $I\subset\mathbb{R}$1 (Szudzik, 2018). These constructions treat “paired functions” not as a relation between two pre-existing functions, but as explicit encodings of pairs into one variable.

4. Reproducing pairs of measurable functions

In frame theory and functional analysis, a reproducing pair is a pair $I\subset\mathbb{R}$2 of weakly measurable functions from a measure space $I\subset\mathbb{R}$3 into a Hilbert space $I\subset\mathbb{R}$4 such that the sesquilinear form

$I\subset\mathbb{R}$5

is well defined and bounded, and the associated operator

$I\subset\mathbb{R}$6

belongs to $I\subset\mathbb{R}$7 (Antoine et al., 2015). When $I\subset\mathbb{R}$8, this reduces to the standard definition of a continuous frame (Antoine et al., 2015).

A central theorem states that each reproducing pair generates two Hilbert spaces, $I\subset\mathbb{R}$9 and $g(x+y)=g(x)g(y)-f(x)f(y).$0, that are conjugate dual to each other with respect to

$g(x+y)=g(x)g(y)-f(x)f(y).$1

(Antoine et al., 2015). The construction proceeds through spaces of measurable coefficient functions $g(x+y)=g(x)g(y)-f(x)f(y).$2 for which the map

$g(x+y)=g(x)g(y)-f(x)f(y).$3

defines a bounded conjugate linear functional, and similarly for $g(x+y)=g(x)g(y)-f(x)f(y).$4 (Antoine et al., 2015). The associated synthesis map is

$g(x+y)=g(x)g(y)-f(x)f(y).$5

The later extension to partial inner product spaces treats the case in which the measurable functions take values in a PIP-space rather than strictly in a Hilbert space (Antoine et al., 2016). The paper states that the reproducing-pair machinery lifts to lattices or scales of Banach or Hilbert spaces, including examples built from weighted sequence spaces, $g(x+y)=g(x)g(y)-f(x)f(y).$6-lattices, and Sobolev scales (Antoine et al., 2016). In this setting, reproducing pairs generalize continuous frames while allowing the analysis operators to land in spaces more general than $g(x+y)=g(x)g(y)-f(x)f(y).$7 (Antoine et al., 2016). The same source emphasizes that $g(x+y)=g(x)g(y)-f(x)f(y).$8 need not be self-adjoint or positive (Antoine et al., 2016).

5. Paired Boolean functions on the slice

A highly specific meaning of “paired functions” appears in the Boolean analysis of the slice $g(x+y)=g(x)g(y)-f(x)f(y).$9, also described as the Johnson scheme (Kiermaier et al., 3 Oct 2025). For $(f,g)$0, the basic function $(f,g)$1 is the characteristic function of the set of all $(f,g)$2 with $(f,g)$3 (Kiermaier et al., 3 Oct 2025). For disjoint $(f,g)$4, the paired function is defined by

$(f,g)$5

The paper states that paired functions are Boolean except for the trivial case $(f,g)$6, when the function is identically $(f,g)$7 (Kiermaier et al., 3 Oct 2025).

The main theorem determines the exact degree of every paired function in terms of $(f,g)$8 and $(f,g)$9: $g$00 The proof is described as elementary and as not involving any spectral methods (Kiermaier et al., 3 Oct 2025).

This construction is noteworthy because, in certain cases, the degree is strictly smaller than the elementary upper bound for the sum of functions (Kiermaier et al., 3 Oct 2025). The paper therefore identifies paired functions as good candidates for fixed-degree Boolean functions of small support size. In the middle layer $g$01, for even degree $g$02, paired functions provide the smallest known non-zero Boolean functions, surpassing the $g$03-pencils; for $g$04 with $g$05, the support size is

$g$06

(Kiermaier et al., 3 Oct 2025). A plausible implication is that the paired construction isolates a cancellation mechanism that is invisible from the naive degree bound on a sum of basic functions.

6. Paired functional data and nonparametric inference

In statistics, “pairs of functions” arise when each observational unit contributes two related curves. One paper develops sign and signed rank tests for paired functional data, where the basic object is the subject-level difference function

$g$07

evaluated on a grid $g$08 (Meyer, 29 Sep 2025). The functional sign test uses

$g$09

and the sufficient-statistic summary

$g$10

or, with zeros allowed, a weighted version $g$11 (Meyer, 29 Sep 2025). After subjectwise summarization, the univariate sign statistic

$g$12

has $g$13 null law under $g$14 (Meyer, 29 Sep 2025). The signed doubly ranked test instead summarizes

$g$15

and then applies a Wilcoxon signed rank statistic

$g$16

The simulation study varied sample size $g$17, grid size $g$18, Gaussian and $g$19-processes, and paired-observation correlations $g$20 (Meyer, 29 Sep 2025). The reported findings are that the signed doubly ranked test consistently maintained nominal type I error and was more powerful than either functional sign test (Meyer, 29 Sep 2025). In the randomized crossover heart-health study, with 34 subjects having sufficient data for both conditions, the signed doubly ranked test gave $g$21 for increased heart rate under the flight condition, while both sign tests gave $g$22 (Meyer, 29 Sep 2025).

A complementary model-based approach addresses sparsely observed paired functional data through a reduced-rank mixed-effects model

$g$23

with the association of the two functional variables modeled through the association of the principal component scores (Zhou et al., 2022). The score vectors and measurement errors are modeled by multivariate scale mixtures of normals, the mean and principal component functions are represented by splines with roughness penalties, and fitting is performed by an EM algorithm (Zhou et al., 2022). The simulation study is reported to show that the proposed method outperforms an existing method that is not designed for robust estimation, and the Type Ia supernovae application records a lower cross-validation error for the robust method in the I-band light curves, $g$24 versus $g$25 for the normal model (Zhou et al., 2022). In this statistical sense, paired functions are not a new function class but a data structure together with inferential procedures that exploit within-subject dependence.

7. Pairing functions in many-body physics

In fermionic many-body physics, the pairing function is the central object in BCS-type paired states. For fully gapped complex paired superfluids in two dimensions, one paper studies the entanglement spectrum using the BCS form of the ground-state wavefunction on a cylinder, with the pairing function

$g$26

and model pairing functions such as $g$27 (Dubail et al., 2011). In the $g$28 model case $g$29, the low-lying entanglement spectrum consists of a single chiral Majorana fermion mode per allowed $g$30, with pseudo-energy

$g$31

and pseudo-Hamiltonian

$g$32

(Dubail et al., 2011). More generally, in the weak-pairing phase of $g$33-wave paired spinless fermions with odd $g$34, the universal low-lying part of the entanglement spectrum consists of $g$35 chiral Majorana fermion modes, while spin-singlet even-$g$36 states yield $g$37 modes (Dubail et al., 2011). The same paper states that the entanglement gap is infinite for the model pairing functions and diverges logarithmically as the model pairing function is approached (Dubail et al., 2011).

A closely related construction appears in paired composite-fermion wave functions for the second Landau level (Henderson et al., 2023). The BCS ansatz is written as

$g$38

and the paired composite-fermion trial state on the sphere is

$g$39

(Henderson et al., 2023). The pairing channels $g$40 are associated respectively with the Pfaffian, anti-Pfaffian, and PH-Pfaffian phases (Henderson et al., 2023). Energy minimization over the coefficients $g$41 reduces the $g$42 energy substantially below that of the Moore–Read wave function at small system sizes, improves the $g$43 case only marginally below the Yutushui–Mross trial function, and makes no qualitative difference in the $g$44 channel, where the wave functions remain energetically unfavorable and show no sign of emergent composite-fermion pairing (Henderson et al., 2023). The effective pairing for $g$45 and $g$46 is reported to be well approximated by a weak-pairing BCS ansatz (Henderson et al., 2023).

Across these physical applications, the pairing function is not merely an auxiliary parametrization. It controls the topological content of the state, the number and type of chiral Majorana modes in the entanglement spectrum, and the quality of variational descriptions of candidate quantum Hall phases (Dubail et al., 2011, Henderson et al., 2023).