Degenerate Sheffer Polynomials

• Degenerate Sheffer polynomials are sequences defined via a degenerate exponential that interpolate classical Bernoulli and Euler polynomials as the parameter vanishes.
• They integrate deterministic and probabilistic methods by incorporating random variable expectations to establish reproducing identities and convolution formulas.
• Their formulation within the λ-umbral calculus framework provides operator tools to derive recurrences, addition formulas, and new hybrid identities.

Searching arXiv for the cited paper and closely related work on degenerate Sheffer and λ-umbral calculus. arxiv_search(query="degenerate Sheffer polynomials lambda umbral calculus Kim Kim", max_results=10) arxiv_search(query="(Kim et al., 27 Jul 2025) Degenerate Sheffer-type polynomials and degenerate Sheffer polynomials associated with a random variable", max_results=5) Degenerate Sheffer polynomials are polynomial sequences built from the degenerate exponential

$$e_\lambda^x(t)=\sum_{n=0}^\infty (x)_{n,\lambda}\frac{t^n}{n!}, \qquad (x)_{n,\lambda}=x(x-\lambda)\cdots(x-(n-1)\lambda),$$

with $\lambda\neq 0$, and they interpolate classical structures as $\lambda\to 0$. In the formulation developed by Kim and Kim, the subject has two linked components: the degenerate Sheffer-type polynomials $T_{n,\lambda}^{(a,b)}(x)$, which hybridize higher-order degenerate Bernoulli and Euler polynomials, and the degenerate Sheffer polynomials $S_{n,\lambda}^Y(x)$ associated with a random variable $Y$, defined through the reciprocal of an expectation involving $e_\lambda^Y(t)$ (Kim et al., 27 Jul 2025).

1. Degenerate exponential framework and classical degenerate families

The starting point is the degenerate exponential

$$e_\lambda^x(t)=\sum_{n=0}^\infty (x)_{n,\lambda}\frac{t^n}{n!}, \qquad (x)_{0,\lambda}=1,$$

which satisfies $e_\lambda^x(t)\to e^{xt}$ and $(x)_{n,0}=x^n$ as $\lambda\neq 0$0. This framework underlies Carlitz’s degenerate Bernoulli and degenerate Euler polynomials, defined respectively by

$\lambda\neq 0$1

and

$\lambda\neq 0$2

Both families satisfy binomial-type addition formulas: $\lambda\neq 0$3

$\lambda\neq 0$4

They reduce to the ordinary Bernoulli and Euler polynomials in the nondegenerate limit. These formulas establish the algebraic background for the hybrid and probabilistic constructions that follow (Kim et al., 27 Jul 2025).

2. Higher-order degenerate Bernoulli and Euler polynomials

For any real $\lambda\neq 0$5, the higher-order degenerate Bernoulli polynomials are defined by

$\lambda\neq 0$6

and the higher-order degenerate Euler polynomials by

$\lambda\neq 0$7

At order $\lambda\neq 0$8, both collapse to the degenerate lower factorial: $\lambda\neq 0$9

The product formula is

$\lambda\to 0$0

$\lambda\to 0$1

A corollary is the shift-by-lower-factorial expansion

$\lambda\to 0$2

$\lambda\to 0$3

The corresponding difference equations are

$\lambda\to 0$4

and

$\lambda\to 0$5

These formulas are not auxiliary; they are the structural components from which the hybrid family $\lambda\to 0$6 is assembled (Kim et al., 27 Jul 2025).

3. Degenerate Sheffer-type polynomials as Bernoulli–Euler hybrids

For real parameters $\lambda\to 0$7, the degenerate Sheffer-type polynomials $\lambda\to 0$8 are defined by

$\lambda\to 0$9

They are explicitly described as a hybrid of higher-order degenerate Bernoulli polynomials, with index $T_{n,\lambda}^{(a,b)}(x)$0, and higher-order degenerate Euler polynomials, with index $T_{n,\lambda}^{(a,b)}(x)$1.

The addition formula is

$T_{n,\lambda}^{(a,b)}(x)$2

A binomial-type expansion in $T_{n,\lambda}^{(a,b)}(x)$3 follows: $T_{n,\lambda}^{(a,b)}(x)$4

Two dual recurrence forms separate the Bernoulli and Euler factors: $T_{n,\lambda}^{(a,b)}(x)$5

$T_{n,\lambda}^{(a,b)}(x)$6

The forward-difference relation is

$T_{n,\lambda}^{(a,b)}(x)$7

A distinctive identity obtained by comparing the two recurrence descriptions is

$T_{n,\lambda}^{(a,b)}(x)$8

This “doubling” relation is a direct consequence of the hybrid construction rather than a restatement of separate Bernoulli or Euler identities (Kim et al., 27 Jul 2025).

4. Degenerate Sheffer polynomials attached to a random variable

Let $T_{n,\lambda}^{(a,b)}(x)$9 be a real random variable whose moment generating function exists in a neighborhood of the origin. The degenerate Sheffer polynomials associated with $S_{n,\lambda}^Y(x)$0 are defined by

$S_{n,\lambda}^Y(x)$1

This introduces a probabilistic mechanism into the degenerate Sheffer framework: the defining invertible factor is not fixed deterministically, but is given by an expectation determined by the law of $S_{n,\lambda}^Y(x)$2.

Two fundamental properties follow immediately. The first is the reproducing-expectation identity

$S_{n,\lambda}^Y(x)$3

The second is convolution under independent sum: if $S_{n,\lambda}^Y(x)$4 are independent copies, then

$S_{n,\lambda}^Y(x)$5

Taking $S_{n,\lambda}^Y(x)$6 reproduces the binomial-type expansion in $S_{n,\lambda}^Y(x)$7.

These formulas identify the family $S_{n,\lambda}^Y(x)$8 as a degenerate Sheffer system governed by probabilistic input. A plausible implication is that distributional data can be translated into polynomial identities through the single quantity $S_{n,\lambda}^Y(x)$9, but the developed theory in the cited work concentrates on the general formal properties and on two concrete distributions (Kim et al., 27 Jul 2025).

5. Uniform and Bernoulli cases, i.i.d. sums, and new connection formulas

For $Y$0,

$Y$1

Hence

$Y$2

and coefficient comparison yields the closed form

$Y$3

For $Y$4,

$Y$5

so that

$Y$6

Therefore

$Y$7

In this case the probabilistically defined Sheffer family coincides exactly with the degenerate Euler polynomials.

The independent-sum formalism extends to $Y$8, where the $Y$9 are i.i.d. copies. For $e_\lambda^Y(t)$0, one obtains

$e_\lambda^Y(t)$1

the $e_\lambda^Y(t)$2th-order Euler polynomials. For $e_\lambda^Y(t)$3, the closed form of the previous theorem extends via the factor $e_\lambda^Y(t)$4.

The same framework also produces new addition and connection formulas among higher-order degenerate Bernoulli and Euler polynomials, including

$e_\lambda^Y(t)$5

and

$e_\lambda^Y(t)$6

The cited exposition characterizes these identities as genuinely new and as arising only in the degenerate hybrid-Sheffer framework (Kim et al., 27 Jul 2025).

6. Relation to λ-umbral calculus and neighboring degenerate Sheffer families

The term “degenerate Sheffer polynomials” is used in a broader λ-umbral-calculus literature as well. In that setting, one defines the degenerate derivative

$e_\lambda^Y(t)$7

and a sequence $e_\lambda^Y(t)$8 is called a $e_\lambda^Y(t)$9-Sheffer sequence for $$e_\lambda^x(t)=\sum_{n=0}^\infty (x)_{n,\lambda}\frac{t^n}{n!}, \qquad (x)_{0,\lambda}=1,$$0 if it satisfies the corresponding operator conditions, equivalently possessing an exponential generating function of the form

$$e_\lambda^x(t)=\sum_{n=0}^\infty (x)_{n,\lambda}\frac{t^n}{n!}, \qquad (x)_{0,\lambda}=1,$$1

Within this framework, fully degenerate Bell and fully degenerate Dowling polynomials are treated as $$e_\lambda^x(t)=\sum_{n=0}^\infty (x)_{n,\lambda}\frac{t^n}{n!}, \qquad (x)_{0,\lambda}=1,$$2-Sheffer sequences, and their expansions, addition theorems, and recurrences are developed through $$e_\lambda^x(t)=\sum_{n=0}^\infty (x)_{n,\lambda}\frac{t^n}{n!}, \qquad (x)_{0,\lambda}=1,$$3-umbral calculus (Ma et al., 2021). Related work on degenerate Hermite polynomials likewise uses the general $$e_\lambda^x(t)=\sum_{n=0}^\infty (x)_{n,\lambda}\frac{t^n}{n!}, \qquad (x)_{0,\lambda}=1,$$4-Sheffer connection formula to represent higher-order degenerate Bernoulli, Euler, and Frobenius–Euler polynomials in a Hermite basis and conversely (Kim et al., 2020).

This broader context clarifies an important terminological point. In (Kim et al., 27 Jul 2025), “degenerate Sheffer-type polynomials” denotes the specific hybrid family

$$e_\lambda^x(t)=\sum_{n=0}^\infty (x)_{n,\lambda}\frac{t^n}{n!}, \qquad (x)_{0,\lambda}=1,$$5

whereas “degenerate Sheffer polynomials associated with $$e_\lambda^x(t)=\sum_{n=0}^\infty (x)_{n,\lambda}\frac{t^n}{n!}, \qquad (x)_{0,\lambda}=1,$$6” denotes the probabilistic family

$$e_\lambda^x(t)=\sum_{n=0}^\infty (x)_{n,\lambda}\frac{t^n}{n!}, \qquad (x)_{0,\lambda}=1,$$7

In related λ-umbral-calculus papers, by contrast, “degenerate Sheffer” or “$$e_\lambda^x(t)=\sum_{n=0}^\infty (x)_{n,\lambda}\frac{t^n}{n!}, \qquad (x)_{0,\lambda}=1,$$8-Sheffer” denotes the general class attached to a pair $$e_\lambda^x(t)=\sum_{n=0}^\infty (x)_{n,\lambda}\frac{t^n}{n!}, \qquad (x)_{0,\lambda}=1,$$9. This suggests that the 2025 construction is best understood not as an isolated nomenclature, but as a specialized realization of the larger $e_\lambda^x(t)\to e^{xt}$0-Sheffer paradigm, with Bernoulli–Euler hybridization and random-variable attachment as its distinguishing features.