Probabilistic Degenerate Frobenius–Euler Polynomials

• Probabilistic degenerate Frobenius–Euler polynomials are defined via generating functions that combine degenerate exponentials and the moment generating function of a random variable.
• They serve as a Sheffer sequence basis, enabling the expansion of arbitrary polynomials with explicit formulas using forward differences and probabilistic degenerate Stirling numbers.
• The framework extends to higher-order cases and specializations for distributions like Bernoulli and Poisson, bridging degenerate and nondegenerate polynomial systems.

Probabilistic degenerate Frobenius–Euler polynomials $h_{n,\lambda}(x\mid u;Y)$ are polynomial sequences attached to a random variable $Y$ whose moment generating function exists in a neighborhood of the origin, together with a complex parameter $u\neq1$ and a degeneracy parameter $\lambda\neq0$. In the formulation of Kim–Kim, they are defined by a generating function built from the degenerate exponential and a distribution-dependent probabilistic degenerate exponential, and they serve as a basis for representing arbitrary polynomials. The resulting theory places the family in the Sheffer framework, provides explicit coefficient formulas via forward differences and probabilistic degenerate Stirling numbers, and extends naturally to higher-order counterparts and distribution-specific specializations (Kim et al., 24 Aug 2025).

1. Definition through generating functions

Let $Y$ be a random variable with moment generating function

$$M_Y(t)\;=\;E\bigl[e^{tY}\bigr]\;=\;\sum_{m=0}^\infty \frac{E[Y^m]}{m!}\,t^m,$$

assumed convergent for $|t|<r$. Fix $u\neq1$ and $\lambda\neq0$. The degenerate exponential is

$$e_{\lambda}(t)\;=\;(1+\lambda\,t)^{1/\lambda}, \qquad \lim_{\lambda\to0}e_{\lambda}(t)=e^t.$$

The probabilistic degenerate exponential of $Y$0 is introduced as

$Y$1

The probabilistic degenerate Frobenius–Euler polynomials are then defined by the generating function

$Y$2

Equivalently, with

$Y$3

one has

$Y$4

When $Y$5, the nondegenerate case is recovered: $Y$6

2. Sheffer-sequence structure and expansion of arbitrary polynomials

A central structural fact is that the sequence $Y$7 is a Sheffer sequence. Specifically, the pair

$Y$8

makes $Y$9 into a Sheffer sequence for $u\neq1$0 (Kim et al., 24 Aug 2025).

This identification yields an expansion theorem for arbitrary polynomials. If $u\neq1$1 has degree $u\neq1$2, then there is a unique representation

$u\neq1$3

The coefficients $u\neq1$4 admit several closed-form expressions. One convenient form uses the forward-difference operator

$u\neq1$5

For $u\neq1$6,

$u\neq1$7

An equivalent form expresses the coefficients through the probabilistic degenerate Stirling numbers of the first kind: $u\neq1$8

Another expression, also recorded in the source, uses the derivatives $u\neq1$9 and the second-kind numbers $\lambda\neq0$0. In this form, the role of the probabilistic degenerate Stirling numbers is to mediate between ordinary polynomial data and the basis $\lambda\neq0$1.

3. Recurrence, difference, and limiting relations

The Sheffer characterization leads directly to operational identities. One has

$\lambda\neq0$2

together with

$\lambda\neq0$3

A notable specialization occurs at $\lambda\neq0$4, where the forward-difference relation becomes

$\lambda\neq0$5

The source also records a formal differential operator formula: $\lambda\neq0$6

The degenerate family connects continuously to the nondegenerate probabilistic Frobenius–Euler polynomials through

$\lambda\neq0$7

This limiting relation places the degenerate theory as a deformation of the probabilistic Frobenius–Euler system rather than as an unrelated construction (Kim et al., 24 Aug 2025).

4. Higher-order probabilistic degenerate Frobenius–Euler polynomials

For an integer $\lambda\neq0$8, the order-$\lambda\neq0$9 polynomials are defined by

$Y$0

The case $Y$1 recovers the degenerate falling factorial: $Y$2

Using the binomial expansion

$Y$3

together with the known expansion of $Y$4, one obtains the explicit formula

$Y$5

where

$Y$6

An equivalent reformulation may be written using the ordinary Stirling numbers $Y$7 to rewrite $Y$8. This higher-order layer extends the basic sequence without altering the dependence on the law of $Y$9, which remains encoded through $$M_Y(t)\;=\;E\bigl[e^{tY}\bigr]\;=\;\sum_{m=0}^\infty \frac{E[Y^m]}{m!}\,t^m,$$0 and the associated probabilistic Stirling numbers.

5. Specializations for Bernoulli and Poisson laws

Concrete formulas emerge once the probabilistic Stirling numbers $$M_Y(t)\;=\;E\bigl[e^{tY}\bigr]\;=\;\sum_{m=0}^\infty \frac{E[Y^m]}{m!}\,t^m,$$1 and $$M_Y(t)\;=\;E\bigl[e^{tY}\bigr]\;=\;\sum_{m=0}^\infty \frac{E[Y^m]}{m!}\,t^m,$$2 are computed from the distribution of $$M_Y(t)\;=\;E\bigl[e^{tY}\bigr]\;=\;\sum_{m=0}^\infty \frac{E[Y^m]}{m!}\,t^m,$$3 (Kim et al., 24 Aug 2025).

For the Bernoulli law

$$M_Y(t)\;=\;E\bigl[e^{tY}\bigr]\;=\;\sum_{m=0}^\infty \frac{E[Y^m]}{m!}\,t^m,$$4

one has

$$M_Y(t)\;=\;E\bigl[e^{tY}\bigr]\;=\;\sum_{m=0}^\infty \frac{E[Y^m]}{m!}\,t^m,$$5

where $$M_Y(t)\;=\;E\bigl[e^{tY}\bigr]\;=\;\sum_{m=0}^\infty \frac{E[Y^m]}{m!}\,t^m,$$6 and $$M_Y(t)\;=\;E\bigl[e^{tY}\bigr]\;=\;\sum_{m=0}^\infty \frac{E[Y^m]}{m!}\,t^m,$$7 are the ordinary degenerate Stirling numbers. Consequently,

$$M_Y(t)\;=\;E\bigl[e^{tY}\bigr]\;=\;\sum_{m=0}^\infty \frac{E[Y^m]}{m!}\,t^m,$$8

The account further states that only $$M_Y(t)\;=\;E\bigl[e^{tY}\bigr]\;=\;\sum_{m=0}^\infty \frac{E[Y^m]}{m!}\,t^m,$$9 survive in the Bernoulli case.

For the Poisson law

$|t|<r$0

one finds

$|t|<r$1

Hence

$|t|<r$2

Analogous closed-form expansions hold for geometric, exponential, normal, and other laws once the corresponding probabilistic Stirling numbers are known. In each case, one works out $|t|<r$3 from the moment generating function of $|t|<r$4.

6. Methodological setting and mathematical significance

The stated aim is to represent arbitrary polynomials in terms of probabilistic Frobenius–Euler polynomials associated with $|t|<r$5, probabilistic degenerate Frobenius–Euler polynomials associated with $|t|<r$6, and their higher-order counterparts. The derivations are obtained with the help of umbral calculus, and the full proofs are attributed to the Sheffer-sequence property, inversion and integral-representation arguments, and elementary difference-operator derivations (Kim et al., 24 Aug 2025).

Within this framework, several structural features become prominent. First, the law of $|t|<r$7 enters through the moment generating function and, more specifically, through $|t|<r$8. Second, the basis-expansion problem for arbitrary $|t|<r$9 is reduced to explicit coefficient extraction formulas involving forward differences or probabilistic degenerate Stirling numbers. Third, the order-$u\neq1$0 extension shows that the same machinery persists under powers of the generating factor $u\neq1$1.

This suggests a modular organization of the theory: once the relevant probabilistic Stirling numbers are available for a chosen distribution, the expansion formulas, recurrence relations, and higher-order identities follow in closed form. In that sense, probabilistic degenerate Frobenius–Euler polynomials occupy a distribution-dependent branch of Sheffer-type special polynomial theory, interpolating between degenerate and nondegenerate Frobenius–Euler structures while retaining explicit umbral and difference-operator realizations.