Probabilistic Frobenius-Euler Polynomials

• Probabilistic Frobenius-Euler polynomials are defined by replacing the classical exponential kernel with the moment-generating function of a random variable, extending the traditional theory.
• They inherit the Appell and Sheffer structure, facilitating basis expansions and connecting to probabilistic Stirling numbers via operator identities.
• The framework unites classical, higher-order, and degenerate versions while providing concrete realizations for distributions like Bernoulli, Poisson, and geometric.

Searching arXiv for the core and papers on probabilistic and classical Frobenius–Euler polynomials. Probabilistic Frobenius–Euler polynomials are Frobenius–Euler-type polynomial sequences in which the classical exponential kernel $e^t$ is replaced by a moment-generating-function input associated with a random variable $Y$. In the classical theory, the Frobenius–Euler polynomials of order $a$ are defined by

$$\left(\frac{1-\lambda}{e^t-\lambda}\right)^a e^{xt} = \sum_{n=0}^{\infty} H_n^{(a)}(x\mid \lambda)\frac{t^n}{n!}, \qquad (\lambda\in\mathbb C,\ \lambda\neq 1),$$

and constitute an Appell sequence in the variable $x$ (Kim et al., 2013). The probabilistic version replaces $e^t$ by $E[e^{Yt}]$, yielding polynomial families attached to the law of $Y$ rather than to the deterministic value $1$. This probabilistic construction, developed explicitly in 2025, is naturally situated within the same umbral-calculus and Sheffer-sequence framework that underlies the classical, higher-order, degenerate, and $q$-deformed Frobenius–Euler theories (Kim et al., 24 Aug 2025).

1. Classical Frobenius–Euler framework

The classical Frobenius–Euler polynomials are defined by the generating function

$Y$0

with higher-order generalization

$Y$1

(Kim et al., 2012). In the notation of higher-order Frobenius–Euler theory, the value at $Y$2 gives the Frobenius–Euler numbers $Y$3 (Kim et al., 2013).

The basic Appell expansion is

$Y$4

equivalently,

$Y$5

with the usual umbral convention replacing powers of $Y$6 by indexed numbers (Kim et al., 2013). The same Appell structure implies the differential relation

$Y$7

a consequence of the general Sheffer/Appell formalism with $Y$8 (Kim et al., 2013). The corresponding addition law

$Y$9

is encoded by the factor $a$0 in the generating function (Kim et al., 2013).

A central operator identity in the basis-theoretic treatment is

$a$1

together with its higher-order analogue

$a$2

(Kim et al., 2012). These identities make the Frobenius–Euler sequence a natural basis for polynomial expansions and also anticipate later order-lowering formulas in the probabilistic theory.

The classical literature further emphasizes that higher-order Frobenius–Euler numbers satisfy the convolution formula

$a$3

which is precisely the coefficient behavior expected when a generating function is raised to an $a$4-th power (Kim et al., 2012). This suggests a sum-of-components interpretation, although the classical papers themselves are not probabilistic.

2. Umbral and Sheffer structure

The modern algebraic treatment of Frobenius–Euler polynomials is based on umbral calculus. If

$a$5

then the associated linear-functional pairing is

$a$6

and

$a$7

(Kim et al., 2013). In this language, Frobenius–Euler polynomials are an Appell sequence, since

$a$8

(Kim et al., 2013).

The broader Sheffer formalism states that a polynomial sequence $a$9 satisfies

$$\left(\frac{1-\lambda}{e^t-\lambda}\right)^a e^{xt} = \sum_{n=0}^{\infty} H_n^{(a)}(x\mid \lambda)\frac{t^n}{n!}, \qquad (\lambda\in\mathbb C,\ \lambda\neq 1),$$0

and has generating function

$$\left(\frac{1-\lambda}{e^t-\lambda}\right)^a e^{xt} = \sum_{n=0}^{\infty} H_n^{(a)}(x\mid \lambda)\frac{t^n}{n!}, \qquad (\lambda\in\mathbb C,\ \lambda\neq 1),$$1

where $$\left(\frac{1-\lambda}{e^t-\lambda}\right)^a e^{xt} = \sum_{n=0}^{\infty} H_n^{(a)}(x\mid \lambda)\frac{t^n}{n!}, \qquad (\lambda\in\mathbb C,\ \lambda\neq 1),$$2 is the compositional inverse of $$\left(\frac{1-\lambda}{e^t-\lambda}\right)^a e^{xt} = \sum_{n=0}^{\infty} H_n^{(a)}(x\mid \lambda)\frac{t^n}{n!}, \qquad (\lambda\in\mathbb C,\ \lambda\neq 1),$$3 (Kim et al., 2021). For the degenerate Frobenius–Euler family this framework becomes

$$\left(\frac{1-\lambda}{e^t-\lambda}\right)^a e^{xt} = \sum_{n=0}^{\infty} H_n^{(a)}(x\mid \lambda)\frac{t^n}{n!}, \qquad (\lambda\in\mathbb C,\ \lambda\neq 1),$$4

and for the higher-order degenerate family

$$\left(\frac{1-\lambda}{e^t-\lambda}\right)^a e^{xt} = \sum_{n=0}^{\infty} H_n^{(a)}(x\mid \lambda)\frac{t^n}{n!}, \qquad (\lambda\in\mathbb C,\ \lambda\neq 1),$$5

(Kim et al., 2021). The paper explicitly derives

$$\left(\frac{1-\lambda}{e^t-\lambda}\right)^a e^{xt} = \sum_{n=0}^{\infty} H_n^{(a)}(x\mid \lambda)\frac{t^n}{n!}, \qquad (\lambda\in\mathbb C,\ \lambda\neq 1),$$6

and

$$\left(\frac{1-\lambda}{e^t-\lambda}\right)^a e^{xt} = \sum_{n=0}^{\infty} H_n^{(a)}(x\mid \lambda)\frac{t^n}{n!}, \qquad (\lambda\in\mathbb C,\ \lambda\neq 1),$$7

(Kim et al., 2021).

This operator-theoretic viewpoint is essential for the later probabilistic theory. It supplies the exact language in which replacing $$\left(\frac{1-\lambda}{e^t-\lambda}\right)^a e^{xt} = \sum_{n=0}^{\infty} H_n^{(a)}(x\mid \lambda)\frac{t^n}{n!}, \qquad (\lambda\in\mathbb C,\ \lambda\neq 1),$$8 by $$\left(\frac{1-\lambda}{e^t-\lambda}\right)^a e^{xt} = \sum_{n=0}^{\infty} H_n^{(a)}(x\mid \lambda)\frac{t^n}{n!}, \qquad (\lambda\in\mathbb C,\ \lambda\neq 1),$$9 becomes meaningful: the probabilistic Frobenius–Euler polynomials are again Sheffer sequences, but now for series determined by the law of $x$0 (Kim et al., 24 Aug 2025).

3. Emergence of the probabilistic theory

The explicit theory of probabilistic Frobenius–Euler polynomials is formulated by taking a random variable $x$1 whose moment generating function exists in a neighborhood of the origin: $x$2 for some $x$3, together with the standing condition

$x$4

(Kim et al., 24 Aug 2025). The notation

$x$5

is introduced in that framework (Kim et al., 24 Aug 2025).

The probabilistic Frobenius–Euler polynomials associated with $x$6 are defined by

$x$7

Their higher-order version is

$x$8

(Kim et al., 24 Aug 2025). In both formulas, the classical factor $x$9 has been replaced by $e^t$0, while the $e^t$1-dependence is carried by $e^t$2.

The classical theory is recovered when $e^t$3, because then

$e^t$4

and the generating function reduces to

$e^t$5

(Kim et al., 24 Aug 2025). This identifies the probabilistic family as a genuine generalization rather than a separate construction.

A plausible implication is that the “probabilistic” modifier does not refer merely to notation but to a structural substitution principle: every place where the classical theory uses the deterministic exponential $e^t$6, the probabilistic theory uses the mgf of $e^t$7. The 2025 paper makes this replacement systematic for the ordinary, higher-order, degenerate, and higher-order degenerate Frobenius–Euler families (Kim et al., 24 Aug 2025).

4. Degenerate and higher-order probabilistic variants

The degenerate exponential is

$e^t$8

with degenerate falling factorial

$e^t$9

(Kim et al., 24 Aug 2025). The probabilistic degenerate Frobenius–Euler polynomials are then defined by

$E[e^{Yt}]$0

and their higher-order version by

$E[e^{Yt}]$1

(Kim et al., 24 Aug 2025).

These probabilistic degenerate families reduce to their nondegenerate analogues as $E[e^{Yt}]$2: $E[e^{Yt}]$3 (Kim et al., 24 Aug 2025). They also reduce to the classical degenerate Frobenius–Euler polynomials when $E[e^{Yt}]$4, because the deterministic degenerate generating function is

$E[e^{Yt}]$5

(Kim et al., 24 Aug 2025).

The nonprobabilistic degenerate literature had already established the same formal pattern: $E[e^{Yt}]$6 together with the higher-order form

$E[e^{Yt}]$7

(Kim et al., 2015). The probabilistic theory therefore extends an already existing hierarchy rather than creating an isolated new family.

5. Representation theory and probabilistic Stirling numbers

A central achievement of the probabilistic theory is the representation of arbitrary polynomials in probabilistic Frobenius–Euler bases. The relevant combinatorial coefficients are the probabilistic Stirling numbers of the second kind

$E[e^{Yt}]$8

and of the first kind

$E[e^{Yt}]$9

with analogous degenerate versions

$Y$0

$Y$1

(Kim et al., 24 Aug 2025).

These satisfy inversion relations: $Y$2 and similarly for the degenerate pair $Y$3 (Kim et al., 24 Aug 2025). This invertibility is indispensable for basis decomposition.

If

$Y$4

then the coefficients are given by

$Y$5

equivalently,

$Y$6

(Kim et al., 24 Aug 2025). The degenerate probabilistic family has the same form with $Y$7 replaced by $Y$8 (Kim et al., 24 Aug 2025).

This extension mirrors the classical basis formulas. In the nonprobabilistic setting, if

$Y$9

then

$1$0

(Kim et al., 2012). The probabilistic formulas retain the same endpoint-difference logic but mediate it through probabilistic Stirling transforms. This suggests that the 2025 theory is best viewed as a probabilistic Shefferization of the older Frobenius–Euler basis theory.

6. Shift, lowering, and order-lowering identities

The probabilistic families inherit Sheffer-type lowering operators. The 2025 paper identifies

$1$1

(Kim et al., 24 Aug 2025). Consequently,

$1$2

with the appropriate choice of $1$3 (Kim et al., 24 Aug 2025).

For the higher-order probabilistic family,

$1$4

equivalently,

$1$5

(Kim et al., 24 Aug 2025). The degenerate higher-order family satisfies the parallel identity

$1$6

(Kim et al., 24 Aug 2025).

These formulas continue a pattern already present in the nonprobabilistic degenerate theory, where

$1$7

(Kim et al., 2015). A plausible implication is that order $1$8 behaves like a convolution parameter in all variants of Frobenius–Euler theory, because the same order-lowering operator recurs throughout the classical, degenerate, and probabilistic settings.

The shift identities also yield normalized endpoint relations. In the probabilistic case,

$1$9

and

$q$0

(Kim et al., 24 Aug 2025). These are the probabilistic analogues of the classical Frobenius–Euler endpoint identity.

7. Concrete probabilistic realizations

The probabilistic theory becomes explicit once $q$1 is specified. The 2025 paper works out examples for Bernoulli, Poisson, geometric, and exponential random variables (Kim et al., 24 Aug 2025).

If $q$2 is Bernoulli with

$q$3

then

$q$4

and the paper states

$q$5

(Kim et al., 24 Aug 2025). Thus, in the Bernoulli case, the probabilistic Stirling numbers become scaled versions of the classical ones.

If $q$6, then

$q$7

and

$q$8

(Kim et al., 24 Aug 2025). This example shows that compound Stirling structures arise naturally when the mgf itself is exponential in $q$9.

If $Y$00 is geometric with parameter $Y$01, then

$Y$02

and

$Y$03

with the degenerate analogue obtained by replacing $Y$04 by $Y$05 (Kim et al., 24 Aug 2025). If $Y$06 is exponential with parameter $Y$07, then

$Y$08

and the paper gives explicit formulas for $Y$09 and $Y$10 in terms of $Y$11 (Kim et al., 24 Aug 2025).

These examples confirm that the probabilistic Frobenius–Euler theory is not purely formal: once the mgf is known, the associated basis coefficients can be made concrete.

8. Earlier probabilistic hints and related constructions

Before the explicit 2025 formulation, several papers contained “probabilistic hints” without developing a full theory. The classical generating function

$Y$12

was repeatedly identified as “mgf-like,” and it was suggested that one could define a formal random variable $Y$13 by

$Y$14

so that

$Y$15

at least formally (Kim et al., 2013). The same source emphasizes that the higher-order parameter $Y$16 would then correspond to convolution power, since powers of a generating function correspond to iterated convolution of coefficient arrays (Kim et al., 2013).

A more concrete probabilistic representation appears in the Fourier-transform paper on the Frobenius–Euler function. For $Y$17,

$Y$18

so this factor is the mgf of a random variable taking values $Y$19, $Y$20, with probabilities

$Y$21

In that parameter range,

$Y$22

with $Y$23 geometric on $Y$24 and

$Y$25

(Araci et al., 2012). This is not the full 2025 probabilistic theory, but it shows that under suitable parameter restrictions the classical Frobenius–Euler kernel already admits a genuine probability interpretation.

A different line of evidence comes from the probabilistic treatment of generalized Euler polynomials. That paper develops Appell polynomials associated with Bernoulli and uniform random variables and notes that the generalized Euler kernel

$Y$26

is exactly the higher-order Frobenius–Euler kernel at $Y$27,

$Y$28

(Ta, 2013). This identifies generalized Euler polynomials as a distinguished probabilistic Frobenius–Euler subfamily.

The 2025 paper may therefore be seen as the culmination of several earlier strands: mgf-like interpretation in classical generating functions, explicit probabilistic realizations for special parameter ranges, and Appell-polynomial methodology imported from probabilistic Bernoulli and Euler theories.

9. Misconceptions and limitations

A frequent misconception is that all Frobenius–Euler polynomials are already probabilistic in the literal sense of arising from a bona fide probability distribution. The literature is explicit that this is not generally so. Multiple papers state that the classical Frobenius–Euler theory is not probabilistic per se: there are no random variables, no expectation formulas, no mgf or pgf interpretations, and no measure-theoretic constructions in those works (Kim et al., 2013, Ilyuta, 2023, Kim et al., 2015).

Even when the generating function has an mgf-like form, admissibility depends on parameters. The Fourier paper notes that positivity fails outside $Y$29, and outside that range one typically obtains a signed or complex measure interpretation rather than an ordinary probability law (Araci et al., 2012). The probabilistic 2025 theory avoids this issue by starting from an actual random variable $Y$30 with existing mgf and then building the Frobenius–Euler family from $Y$31 (Kim et al., 24 Aug 2025).

Another misconception is that “orthogonality-type properties” in $Y$32-Frobenius–Euler work imply orthogonality with respect to a positive measure. The relevant $Y$33-paper makes clear that these are dual-basis identities in $Y$34-umbral algebra, not Hilbert-space orthogonality (Kim, 2013). This distinction matters because probabilistic interpretations often tempt a measure-theoretic reading that the source does not support.

10. Position within the broader Frobenius–Euler landscape

Probabilistic Frobenius–Euler polynomials occupy a late and conceptually unifying position in the subject. The classical strand established generating functions, Appell properties, higher-order convolution identities, and basis expansions (Kim et al., 2012, Kim et al., 2013). The degenerate strand introduced $Y$35-based deformations, difference identities, and Stirling-number transforms between ordinary and degenerate families (Kim et al., 2015, Kim et al., 2021). The generalized and Fourier/Dedekind strands showed that Frobenius–Euler objects naturally interact with root-of-unity sums, multiplication formulas, and harmonic-analytic transforms (Ilyuta, 2023, Araci et al., 2012). The $Y$36-strand produced $Y$37-Appell analogues and dual-basis formulas (Kim, 2013).

The probabilistic theory of 2025 adds a new axis: dependence on the law of a random variable $Y$38 through its mgf. Its defining formulas

$Y$39

$Y$40

and their degenerate analogues provide a common framework in which classical, higher-order, and degenerate Frobenius–Euler polynomials appear as special cases (Kim et al., 24 Aug 2025).

This suggests that “probabilistic Frobenius–Euler polynomials” are best understood not as a single special sequence but as a law-dependent Sheffer-Appell class. Their structure is controlled simultaneously by the Frobenius parameter $Y$41, the order parameter $Y$42, the degeneration parameter $Y$43, and the moment structure of $Y$44. The resulting theory is both a probabilistic generalization of Frobenius–Euler polynomials and a Frobenius–Euler specialization of probabilistic Appell-Sheffer calculus.