Degenerate Sheffer-Type Polynomials

• Degenerate Sheffer-type polynomials are sequences defined via λ-deformed exponential generating functions, notably replacing e^(xt) with the Carlitz degenerate exponential eₗˣ(t).
• They utilize the λ-umbral calculus framework to construct mixed Bernoulli–Euler systems, probabilistic representations, and smoothly recover classical polynomial systems as λ → 0.
• The framework underpins transformation formulas, explicit connection coefficients, and forward-difference identities that substitute traditional derivative rules with shift operations.

Degenerate Sheffer-type polynomials are polynomial sequences defined by Sheffer-style exponential generating functions in which the classical kernel $e^{xt}$ is replaced by a $\lambda$-deformed kernel, most commonly the Carlitz degenerate exponential

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

or by analogous logarithmic/exponential deformations in an ordinary Sheffer framework. In the modern literature, they are treated through $\lambda$-umbral calculus, where the Sheffer pair $(g_\lambda,f_\lambda)$ and the degenerate falling-factorial basis $\{(x)_{n,\lambda}\}_{n\ge0}$ replace the classical Appell–Sheffer apparatus based on ordinary powers and derivatives. This framework supports mixed Bernoulli–Euler constructions, probabilistic Appell-type sequences associated with random variables, representation theory between different degenerate families, and systematic $\lambda\to0$ recovery of classical Bernoulli, Euler, Bell, Hermite, poly-Bernoulli, and related polynomial systems (Kim et al., 27 Jul 2025, Kim et al., 2020, Ma et al., 2021, Kim et al., 2013).

1. Foundational framework

The standard analytic ingredients are the degenerate falling factorial and the degenerate exponential

$$(x)_{0,\lambda}=1,\qquad (x)_{n,\lambda}=x(x-\lambda)(x-2\lambda)\cdots\bigl(x-(n-1)\lambda\bigr),$$

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

They satisfy

$$\lim_{\lambda\to0}e_\lambda^x(t)=e^{xt}, \qquad \lim_{\lambda\to0}(x)_{n,\lambda}=x^n,$$

and the binomial-type identity

$\lambda$0

These relations make $\lambda$1 the natural basic sequence for degenerate Appell- and Sheffer-type constructions (Kim et al., 27 Jul 2025).

In the $\lambda$2-umbral formalism, a $\lambda$3-Sheffer sequence $\lambda$4 is characterized by the bracket identity

$\lambda$5

and by the generating form

$\lambda$6

where $\lambda$7 is the compositional inverse of $\lambda$8. A general connection-coefficient formula expresses one $\lambda$9-Sheffer family in another: $$e_\lambda^x(t)=(1+\lambda t)^{x/\lambda} =\sum_{n=0}^\infty (x)_{n,\lambda}\frac{t^n}{n!}, \qquad (x)_{n,\lambda}=x(x-\lambda)\cdots(x-(n-1)\lambda),$$0 This is the structural mechanism behind many explicit representations in the literature (Kim et al., 2020).

A broader Sheffer perspective predates the degenerate setting. In the ordinary umbral theory, a Sheffer sequence for $$e_\lambda^x(t)=(1+\lambda t)^{x/\lambda} =\sum_{n=0}^\infty (x)_{n,\lambda}\frac{t^n}{n!}, \qquad (x)_{n,\lambda}=x(x-\lambda)\cdots(x-(n-1)\lambda),$$1 has EGF $$e_\lambda^x(t)=(1+\lambda t)^{x/\lambda} =\sum_{n=0}^\infty (x)_{n,\lambda}\frac{t^n}{n!}, \qquad (x)_{n,\lambda}=x(x-\lambda)\cdots(x-(n-1)\lambda),$$2, obeys the lowering relation $$e_\lambda^x(t)=(1+\lambda t)^{x/\lambda} =\sum_{n=0}^\infty (x)_{n,\lambda}\frac{t^n}{n!}, \qquad (x)_{n,\lambda}=x(x-\lambda)\cdots(x-(n-1)\lambda),$$3, and admits addition formulas determined by the associated sequence of $$e_\lambda^x(t)=(1+\lambda t)^{x/\lambda} =\sum_{n=0}^\infty (x)_{n,\lambda}\frac{t^n}{n!}, \qquad (x)_{n,\lambda}=x(x-\lambda)\cdots(x-(n-1)\lambda),$$4. Degenerate Sheffer-type polynomials inherit these patterns after replacing $$e_\lambda^x(t)=(1+\lambda t)^{x/\lambda} =\sum_{n=0}^\infty (x)_{n,\lambda}\frac{t^n}{n!}, \qquad (x)_{n,\lambda}=x(x-\lambda)\cdots(x-(n-1)\lambda),$$5 by $$e_\lambda^x(t)=(1+\lambda t)^{x/\lambda} =\sum_{n=0}^\infty (x)_{n,\lambda}\frac{t^n}{n!}, \qquad (x)_{n,\lambda}=x(x-\lambda)\cdots(x-(n-1)\lambda),$$6 or, in some variants, by inserting $$e_\lambda^x(t)=(1+\lambda t)^{x/\lambda} =\sum_{n=0}^\infty (x)_{n,\lambda}\frac{t^n}{n!}, \qquad (x)_{n,\lambda}=x(x-\lambda)\cdots(x-(n-1)\lambda),$$7-deformed logarithmic compositions into otherwise classical Sheffer data (Kim et al., 2013, Kim, 2017).

2. Canonical families and Sheffer data

Several well-studied degenerate polynomial families fit naturally into Sheffer-type schemes. Some are genuine $$e_\lambda^x(t)=(1+\lambda t)^{x/\lambda} =\sum_{n=0}^\infty (x)_{n,\lambda}\frac{t^n}{n!}, \qquad (x)_{n,\lambda}=x(x-\lambda)\cdots(x-(n-1)\lambda),$$8-Sheffer sequences with the Carlitz kernel $$e_\lambda^x(t)=(1+\lambda t)^{x/\lambda} =\sum_{n=0}^\infty (x)_{n,\lambda}\frac{t^n}{n!}, \qquad (x)_{n,\lambda}=x(x-\lambda)\cdots(x-(n-1)\lambda),$$9; others are ordinary Sheffer sequences generated by logarithmic deformations.

Family	Generating function or Sheffer data	Source
Higher-order degenerate Bernoulli	$\lambda$0	(Kim et al., 27 Jul 2025)
Higher-order degenerate Euler	$\lambda$1	(Kim et al., 27 Jul 2025)
Degenerate poly-Bernoulli	$\lambda$2; pair $\lambda$3	(2002.04520)
New type degenerate poly-Euler	$\lambda$4; pair $\lambda$5	(Ma et al., 2022)
Degenerate Hermite	$\lambda$6; pair $\lambda$7	(Kim et al., 2020)
Fully degenerate Bell	$\lambda$8, EGF $\lambda$9	(Ma et al., 2021)
Degenerate Cauchy, second kind	ordinary Sheffer pair $(g_\lambda,f_\lambda)$0	(Kim, 2017)

The most classical degenerate Sheffer-type examples are the Carlitz degenerate Bernoulli and Euler families: $(g_\lambda,f_\lambda)$1 Their higher-order versions are obtained by raising the prefactors to a real order parameter $(g_\lambda,f_\lambda)$2. For $(g_\lambda,f_\lambda)$3, both reduce to the basic sequence $(g_\lambda,f_\lambda)$4, and as $(g_\lambda,f_\lambda)$5 they recover the higher-order classical Bernoulli and Euler polynomials (Kim et al., 27 Jul 2025).

The polylogarithmic branch of the theory introduces additional Sheffer-type families. Degenerate poly-Bernoulli polynomials are defined through the degenerate polylogarithm $(g_\lambda,f_\lambda)$6 and have Sheffer pair

$(g_\lambda,f_\lambda)$7

while the new type degenerate poly-Euler polynomials use

$(g_\lambda,f_\lambda)$8

In both cases the x-dependence is expanded in the $(g_\lambda,f_\lambda)$9-falling-factorial basis, so the resulting polynomials remain Appell-type in the degenerate umbral sense (2002.04520, Ma et al., 2022).

A useful structural distinction is that not every “degenerate” family belongs to the same kernel-based category. Fully degenerate Bell and Dowling polynomials are built inside $\{(x)_{n,\lambda}\}_{n\ge0}$0-umbral calculus through $\{(x)_{n,\lambda}\}_{n\ge0}$1 and $\{(x)_{n,\lambda}\}_{n\ge0}$2, whereas the degenerate Cauchy polynomials of the second kind are ordinary Sheffer polynomials whose deformation comes from the composite map $\{(x)_{n,\lambda}\}_{n\ge0}$3. This difference is explicit in their Sheffer pairs and in the operators governing their lowering laws (Ma et al., 2021, Kim, 2017).

3. Mixed Bernoulli–Euler families and the degenerate Sheffer-type polynomials $\{(x)_{n,\lambda}\}_{n\ge0}$4

A central recent construction is the two-parameter mixed family

$\{(x)_{n,\lambda}\}_{n\ge0}$5

where $\{(x)_{n,\lambda}\}_{n\ge0}$6. This family hybridizes the higher-order degenerate Bernoulli and Euler factors and is Sheffer, in the degenerate umbral sense, for

$\{(x)_{n,\lambda}\}_{n\ge0}$7

Because $\{(x)_{n,\lambda}\}_{n\ge0}$8, the family is Appell-type within the degenerate setting, with forward differences replacing ordinary derivatives (Kim et al., 27 Jul 2025).

Two basic expansions control the family: $\{(x)_{n,\lambda}\}_{n\ge0}$9

$\lambda\to0$0

The coefficients at $\lambda\to0$1 are precisely the EGF coefficients of $\lambda\to0$2. In particular,

$\lambda\to0$3

and the first polynomial is

$\lambda\to0$4

The cases $\lambda\to0$5, $\lambda\to0$6, and $\lambda\to0$7 recover, respectively, $\lambda\to0$8, $\lambda\to0$9, and $$(x)_{0,\lambda}=1,\qquad (x)_{n,\lambda}=x(x-\lambda)(x-2\lambda)\cdots\bigl(x-(n-1)\lambda\bigr),$$0 (Kim et al., 27 Jul 2025).

The principal operational identity is the forward-difference law

$$(x)_{0,\lambda}=1,\qquad (x)_{n,\lambda}=x(x-\lambda)(x-2\lambda)\cdots\bigl(x-(n-1)\lambda\bigr),$$1

This is the mixed analogue of the Appell derivative rule. A second key relation is the averaging identity

$$(x)_{0,\lambda}=1,\qquad (x)_{n,\lambda}=x(x-\lambda)(x-2\lambda)\cdots\bigl(x-(n-1)\lambda\bigr),$$2

equivalently

$$(x)_{0,\lambda}=1,\qquad (x)_{n,\lambda}=x(x-\lambda)(x-2\lambda)\cdots\bigl(x-(n-1)\lambda\bigr),$$3

These formulas intertwine the Bernoulli and Euler parts of the mixed system and produce connection identities between $$(x)_{0,\lambda}=1,\qquad (x)_{n,\lambda}=x(x-\lambda)(x-2\lambda)\cdots\bigl(x-(n-1)\lambda\bigr),$$4 and $$(x)_{0,\lambda}=1,\qquad (x)_{n,\lambda}=x(x-\lambda)(x-2\lambda)\cdots\bigl(x-(n-1)\lambda\bigr),$$5 (Kim et al., 27 Jul 2025).

The same framework yields new formulas for the higher-order degenerate Bernoulli and Euler polynomials themselves. Among the fundamental ones are the convolution laws

$$(x)_{0,\lambda}=1,\qquad (x)_{n,\lambda}=x(x-\lambda)(x-2\lambda)\cdots\bigl(x-(n-1)\lambda\bigr),$$6

$$(x)_{0,\lambda}=1,\qquad (x)_{n,\lambda}=x(x-\lambda)(x-2\lambda)\cdots\bigl(x-(n-1)\lambda\bigr),$$7

and the forward-difference identities

$$(x)_{0,\lambda}=1,\qquad (x)_{n,\lambda}=x(x-\lambda)(x-2\lambda)\cdots\bigl(x-(n-1)\lambda\bigr),$$8

$$(x)_{0,\lambda}=1,\qquad (x)_{n,\lambda}=x(x-\lambda)(x-2\lambda)\cdots\bigl(x-(n-1)\lambda\bigr),$$9

A particularly characteristic mixed formula is

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

together with the “halving” identity

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

These relations show that the mixed family is not merely a formal product but a device for transporting identities between the Bernoulli and Euler sectors (Kim et al., 27 Jul 2025).

4. Probabilistic degenerate Sheffer polynomials

A probabilistic branch of the theory associates a degenerate Sheffer family to a random variable $$e_\lambda^x(t)=\sum_{k=0}^\infty (x)_{k,\lambda}\frac{t^k}{k!}, \qquad e_\lambda(t)=e_\lambda^1(t)=(1+\lambda t)^{1/\lambda}.$$2 whose classical moment generating function exists in a neighborhood of the origin. The defining EGF is

$$e_\lambda^x(t)=\sum_{k=0}^\infty (x)_{k,\lambda}\frac{t^k}{k!}, \qquad e_\lambda(t)=e_\lambda^1(t)=(1+\lambda t)^{1/\lambda}.$$3

with Sheffer pair

$$e_\lambda^x(t)=\sum_{k=0}^\infty (x)_{k,\lambda}\frac{t^k}{k!}, \qquad e_\lambda(t)=e_\lambda^1(t)=(1+\lambda t)^{1/\lambda}.$$4

This produces a degenerate Appell family attached directly to the distribution of $$e_\lambda^x(t)=\sum_{k=0}^\infty (x)_{k,\lambda}\frac{t^k}{k!}, \qquad e_\lambda(t)=e_\lambda^1(t)=(1+\lambda t)^{1/\lambda}.$$5 (Kim et al., 27 Jul 2025).

The defining expectation identity is

$$e_\lambda^x(t)=\sum_{k=0}^\infty (x)_{k,\lambda}\frac{t^k}{k!}, \qquad e_\lambda(t)=e_\lambda^1(t)=(1+\lambda t)^{1/\lambda}.$$6

Thus the family is characterized by “mean-translation” from a random shift to the degenerate falling factorials. If $$e_\lambda^x(t)=\sum_{k=0}^\infty (x)_{k,\lambda}\frac{t^k}{k!}, \qquad e_\lambda(t)=e_\lambda^1(t)=(1+\lambda t)^{1/\lambda}.$$7 and $$e_\lambda^x(t)=\sum_{k=0}^\infty (x)_{k,\lambda}\frac{t^k}{k!}, \qquad e_\lambda(t)=e_\lambda^1(t)=(1+\lambda t)^{1/\lambda}.$$8 are independent, then

$$e_\lambda^x(t)=\sum_{k=0}^\infty (x)_{k,\lambda}\frac{t^k}{k!}, \qquad e_\lambda(t)=e_\lambda^1(t)=(1+\lambda t)^{1/\lambda}.$$9

so independence becomes convolution at the level of Sheffer coefficients. The triangular expansion

$$\lim_{\lambda\to0}e_\lambda^x(t)=e^{xt}, \qquad \lim_{\lambda\to0}(x)_{n,\lambda}=x^n,$$0

shows that the family is again built over the basis $$\lim_{\lambda\to0}e_\lambda^x(t)=e^{xt}, \qquad \lim_{\lambda\to0}(x)_{n,\lambda}=x^n,$$1 (Kim et al., 27 Jul 2025).

Two distributions are worked out explicitly. For $$\lim_{\lambda\to0}e_\lambda^x(t)=e^{xt}, \qquad \lim_{\lambda\to0}(x)_{n,\lambda}=x^n,$$2,

$$\lim_{\lambda\to0}e_\lambda^x(t)=e^{xt}, \qquad \lim_{\lambda\to0}(x)_{n,\lambda}=x^n,$$3

hence

$$\lim_{\lambda\to0}e_\lambda^x(t)=e^{xt}, \qquad \lim_{\lambda\to0}(x)_{n,\lambda}=x^n,$$4

Consequently,

$$\lim_{\lambda\to0}e_\lambda^x(t)=e^{xt}, \qquad \lim_{\lambda\to0}(x)_{n,\lambda}=x^n,$$5

with initial terms

$$\lim_{\lambda\to0}e_\lambda^x(t)=e^{xt}, \qquad \lim_{\lambda\to0}(x)_{n,\lambda}=x^n,$$6

$$\lim_{\lambda\to0}e_\lambda^x(t)=e^{xt}, \qquad \lim_{\lambda\to0}(x)_{n,\lambda}=x^n,$$7

The paper states that these interpolate toward the classical Bernoulli polynomials as $$\lim_{\lambda\to0}e_\lambda^x(t)=e^{xt}, \qquad \lim_{\lambda\to0}(x)_{n,\lambda}=x^n,$$8 (Kim et al., 27 Jul 2025).

For $$\lim_{\lambda\to0}e_\lambda^x(t)=e^{xt}, \qquad \lim_{\lambda\to0}(x)_{n,\lambda}=x^n,$$9,

$\lambda$00

so

$\lambda$01

and therefore

$\lambda$02

For an $\lambda$03-fold sum of independent $\lambda$04 variables, the associated family becomes exactly the higher-order degenerate Euler polynomial system: $\lambda$05 This identifies classical degenerate Euler theory as a special case of a random-variable construction rather than as an isolated formal family (Kim et al., 27 Jul 2025).

A plausible implication is that the probabilistic construction provides a unifying interpretation of many degenerate Appell-type sequences: distributional averaging determines the prefactor $\lambda$06, while the $\lambda$07-falling-factorial basis controls the polynomial part.

5. Representation theory, transformations, and inter-family expansions

One of the strongest features of the degenerate Sheffer-type framework is its capacity to transfer formulas between different polynomial bases. Degenerate Hermite polynomials furnish a model case. They are defined by

$\lambda$08

with $\lambda$09-Sheffer pair

$\lambda$10

Their explicit formula is

$\lambda$11

and the first values are

$\lambda$12

The general $\lambda$13-Sheffer-to-$\lambda$14-Sheffer connection formula is then used to represent higher-order degenerate Bernoulli, Euler, and Frobenius–Euler polynomials in the Hermite basis, and conversely to expand degenerate Hermite polynomials in those bases (Kim et al., 2020).

The same representation logic appears in the fully degenerate Bell and Dowling setting. The fully degenerate Bell polynomials satisfy

$\lambda$15

and admit the expansion

$\lambda$16

The fully degenerate Dowling polynomials are defined by

$\lambda$17

with EGF

$\lambda$18

Here again, Sheffer structure supplies change-of-basis formulas, expansions of degenerate Bernoulli and falling-factorial sequences, and inversion-type relations between Bell- and Dowling-type objects (Ma et al., 2021).

Series-transformation methods provide another route to Sheffer-type identities. In the degenerate adaptation of Boyadzhiev’s transformation formula, if

$\lambda$19

then

$\lambda$20

This template is specialized to degenerate Stirling numbers, Bell polynomials, Fubini polynomials, and degenerate poly-Bernoulli polynomials. In particular, degenerate Bell polynomials have EGF

$\lambda$21

while degenerate poly-Bernoulli polynomials satisfy

$\lambda$22

These formulas exhibit the same Sheffer-type pattern but connect it directly to series-transformation and Stirling-inversion techniques (Kim et al., 2021).

A common theme across these examples is that Sheffer-type status is not merely classificatory. It governs representation in alternate bases, finite connection coefficients, lowering actions, and convolution identities, and it often turns generating-function multiplication into explicit coefficient formulas.

6. Limits, variants, and structural issues

The universal organizing principle is classical recovery as $\lambda$23. For the mixed Bernoulli–Euler family,

$\lambda$24

and for the basic Carlitz families

$\lambda$25

Degenerate Hermite polynomials satisfy

$\lambda$26

while fully degenerate Bell and Dowling polynomials converge to the ordinary Bell and Dowling polynomials. Degenerate poly-Bernoulli and poly-Euler constructions similarly recover their classical polylogarithmic counterparts in the limit (Kim et al., 27 Jul 2025, Kim et al., 2020, Ma et al., 2021, 2002.04520, Ma et al., 2022).

A persistent structural feature is the replacement of derivative identities by forward-difference or shift relations. For example,

$\lambda$27

$\lambda$28

whereas classical Appell sequences would be described directly by derivatives. This is not a cosmetic reformulation; it reflects the role of multiplication by $\lambda$29 and the basic shift $\lambda$30 in the degenerate kernel (Kim et al., 27 Jul 2025).

A useful correction to a common oversimplification is that “degenerate Sheffer-type” does not denote a single uniform formalism. Much of the literature uses the Carlitz kernel $\lambda$31 and the $\lambda$32-umbral calculus of Kim–Kim, but some families are ordinary Sheffer sequences with deformed logarithmic inputs. The degenerate Cauchy polynomials of the second kind are the clearest example: $\lambda$33 with ordinary Sheffer pair

$\lambda$34

They are therefore degenerate Sheffer-type in the broad encyclopedic sense, but not a $\lambda$35-Sheffer sequence built directly from the Carlitz kernel (Kim, 2017).

Another objective qualification concerns interpretation. The degenerate Hermite paper explicitly recalls the classical orthogonality of ordinary Hermite polynomials but does not derive orthogonality or weight functions for the degenerate Hermite family. More generally, much of the current literature emphasizes generating functions, connection coefficients, and combinatorial identities rather than spectral or measure-theoretic properties (Kim et al., 2020).

Taken together, these developments show that degenerate Sheffer-type polynomials form a coherent but heterogeneous domain. Their unifying core is the Sheffer principle—factorization into an invertible prefactor and a deformed exponential kernel—while their diversity lies in the choice of deformation, the role of Stirling-type transforms, the availability of probabilistic models, and the degree to which classical operator identities survive as forward-difference analogues.