---
title: Degenerate Sheffer Polynomials
url: https://www.emergentmind.com/topics/degenerate-sheffer-polynomials
type: topic
---

# Degenerate Sheffer Polynomials

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="2507.20167 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 \(e_\lambda^Y(t)\) [2507.20167].

## 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 \(\lambda\to 0\). This framework underlies Carlitz’s degenerate Bernoulli and degenerate Euler polynomials, defined respectively by
\[
\left(\frac{t}{e_\lambda(t)-1}\right)e_\lambda^x(t)
=
\sum_{n=0}^\infty \beta_{n,\lambda}(x)\frac{t^n}{n!},
\]
and
\[
\left(\frac{2}{e_\lambda(t)+1}\right)e_\lambda^x(t)
=
\sum_{n=0}^\infty \mathcal{E}_{n,\lambda}(x)\frac{t^n}{n!}.
\]

Both families satisfy binomial-type addition formulas:
\[
\beta_{n,\lambda}(x+y)=\sum_{k=0}^n \binom{n}{k}\beta_{k,\lambda}(x)(y)_{n-k,\lambda},
\]
\[
\mathcal{E}_{n,\lambda}(x+y)=\sum_{k=0}^n \binom{n}{k}\mathcal{E}_{k,\lambda}(x)(y)_{n-k,\lambda}.
\]
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 [2507.20167].

## 2. Higher-order degenerate Bernoulli and Euler polynomials

For any real \(\alpha\), the higher-order degenerate Bernoulli polynomials are defined by
\[
\left(\frac{t}{e_\lambda(t)-1}\right)^\alpha e_\lambda^x(t)
=
\sum_{n=0}^\infty \beta_{n,\lambda}^{(\alpha)}(x)\frac{t^n}{n!},
\]
and the higher-order degenerate Euler polynomials by
\[
\left(\frac{2}{e_\lambda(t)+1}\right)^\alpha e_\lambda^x(t)
=
\sum_{n=0}^\infty \mathcal{E}_{n,\lambda}^{(\alpha)}(x)\frac{t^n}{n!}.
\]
At order \(0\), both collapse to the degenerate lower factorial:
\[
\beta_{n,\lambda}^{(0)}(x)=(x)_{n,\lambda},
\qquad
\mathcal{E}_{n,\lambda}^{(0)}(x)=(x)_{n,\lambda}.
\]

The product formula is
\[
\beta_{n,\lambda}^{(\alpha+\beta)}(x+y)
=
\sum_{k=0}^n
\binom{n}{k}\,
\beta_{k,\lambda}^{(\alpha)}(x)\,
\beta_{n-k,\lambda}^{(\beta)}(y),
\]
\[
\mathcal{E}_{n,\lambda}^{(\alpha+\beta)}(x+y)
=
\sum_{k=0}^n
\binom{n}{k}\,
\mathcal{E}_{k,\lambda}^{(\alpha)}(x)\,
\mathcal{E}_{n-k,\lambda}^{(\beta)}(y).
\]
A corollary is the shift-by-lower-factorial expansion
\[
\beta_{n,\lambda}^{(\alpha)}(x+y)
=
\sum_{k=0}^n
\binom{n}{k}\,
\beta_{k,\lambda}^{(\alpha)}(x)\,(y)_{n-k,\lambda},
\]
\[
\mathcal{E}_{n,\lambda}^{(\alpha)}(x+y)
=
\sum_{k=0}^n
\binom{n}{k}\,
\mathcal{E}_{k,\lambda}^{(\alpha)}(x)\,(y)_{n-k,\lambda}.
\]

The corresponding difference equations are
\[
\beta_{n,\lambda}^{(\alpha)}(x+1)-\beta_{n,\lambda}^{(\alpha)}(x)
=
n\,\beta_{n-1,\lambda}^{(\alpha-1)}(x),
\]
and
\[
\mathcal{E}_{n,\lambda}^{(\alpha)}(x+1)+\mathcal{E}_{n,\lambda}^{(\alpha)}(x)
=
2\,\mathcal{E}_{n,\lambda}^{(\alpha-1)}(x).
\]
These formulas are not auxiliary; they are the structural components from which the hybrid family \(T_{n,\lambda}^{(a,b)}(x)\) is assembled [2507.20167].

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

For real parameters \(a,b\), the degenerate Sheffer-type polynomials \(T_{n,\lambda}^{(a,b)}(x)\) are defined by
\[
\left(\frac{t}{e_\lambda(t)-1}\right)^a
\left(\frac{2}{e_\lambda(t)+1}\right)^b
e_\lambda^x(t)
=
\sum_{n=0}^\infty T_{n,\lambda}^{(a,b)}(x)\frac{t^n}{n!}.
\]
They are explicitly described as a hybrid of higher-order degenerate Bernoulli polynomials, with index \(a\), and higher-order degenerate Euler polynomials, with index \(b\).

The addition formula is
\[
T_{n,\lambda}^{(a,b)}(x+y)
=
\sum_{k=0}^n
\binom{n}{k}\,
\beta_{k,\lambda}^{(a)}(x)\,
\mathcal{E}_{n-k,\lambda}^{(b)}(y).
\]
A binomial-type expansion in \(x\) follows:
\[
T_{n,\lambda}^{(a,b)}(x)
=
\sum_{k=0}^n
\binom{n}{k}\,
T_{k,\lambda}^{(a,b)}(0)\,
(x)_{n-k,\lambda}.
\]

Two dual recurrence forms separate the Bernoulli and Euler factors:
\[
T_{n,\lambda}^{(a,b)}(x)
=
\sum_{k=0}^n
\binom{n}{k}\,
T_{k,\lambda}^{(a-1,b)}(0)\,
\beta_{n-k,\lambda}(x),
\]
\[
T_{n,\lambda}^{(a,b)}(x)
=
\sum_{k=0}^n
\binom{n}{k}\,
T_{k,\lambda}^{(a,b-1)}(0)\,
\mathcal{E}_{n-k,\lambda}(x).
\]
The forward-difference relation is
\[
T_{n,\lambda}^{(a,b)}(x+1)-T_{n,\lambda}^{(a,b)}(x)
=
n\,T_{n-1,\lambda}^{(a-1,b)}(x).
\]

A distinctive identity obtained by comparing the two recurrence descriptions is
\[
\sum_{k=0}^n
\binom{n}{k}\,
\beta_{k,\lambda}(x)\,
\mathcal{E}_{n-k,\lambda}(x)
=
2^n\,\beta_{n,\lambda/2}\!\left(\frac{x}{2}\right).
\]
This “doubling” relation is a direct consequence of the hybrid construction rather than a restatement of separate Bernoulli or Euler identities [2507.20167].

## 4. Degenerate Sheffer polynomials attached to a random variable

Let \(Y\) be a real random variable whose moment generating function exists in a neighborhood of the origin. The degenerate Sheffer polynomials associated with \(Y\) are defined by
\[
\frac{1}{E[e_\lambda^Y(t)]}\,e_\lambda^x(t)
=
\sum_{n=0}^\infty S_{n,\lambda}^Y(x)\frac{t^n}{n!}.
\]
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 \(Y\).

Two fundamental properties follow immediately. The first is the reproducing-expectation identity
\[
E\bigl[S_{n,\lambda}^Y(x+Y)\bigr]=(x)_{n,\lambda}.
\]
The second is convolution under independent sum: if \(Y_1,Y_2\) are independent copies, then
\[
S_{n,\lambda}^{Y_1+Y_2}(x+y)
=
\sum_{k=0}^n
\binom{n}{k}\,
S_{k,\lambda}^{Y_1}(x)\,
S_{n-k,\lambda}^{Y_2}(y).
\]
Taking \(Y_2=0\) reproduces the binomial-type expansion in \((x)_{n,\lambda}\).

These formulas identify the family \(S_{n,\lambda}^Y(x)\) 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 \(E[e_\lambda^Y(t)]\), but the developed theory in the cited work concentrates on the general formal properties and on two concrete distributions [2507.20167].

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

For \(Y\sim \mathrm{Uniform}[0,1]\),
\[
E[e_\lambda^Y(t)]
=
\int_0^1 e_\lambda^y(t)\,dy
=
\left(\frac{\lambda t}{\log(1+\lambda t)}\right)\frac{e_\lambda(t)-1}{t}.
\]
Hence
\[
\sum_{n=0}^\infty S_{n,\lambda}^Y(x)\frac{t^n}{n!}
=
\left(\frac{\log(1+\lambda t)}{\lambda t}\right)
\left(\frac{t}{e_\lambda(t)-1}\right)e_\lambda^x(t),
\]
and coefficient comparison yields the closed form
\[
S_{n,\lambda}^{U[0,1]}(x)
=
\sum_{k=0}^n
\binom{n}{k}\,
\beta_{k,\lambda}(x)\,
\frac{(-\lambda)^{n-k}(n-k)!}{n-k+1}.
\]

For \(Y\sim \mathrm{Bernoulli}(1/2)\),
\[
E[e_\lambda^Y(t)]
=
\frac12+\frac12 e_\lambda(t)
=
\frac{e_\lambda(t)+1}{2},
\]
so that
\[
\sum_{n=0}^\infty S_{n,\lambda}^Y(x)\frac{t^n}{n!}
=
\frac{2}{e_\lambda(t)+1}e_\lambda^x(t)
=
\sum_{n=0}^\infty \mathcal{E}_{n,\lambda}(x)\frac{t^n}{n!}.
\]
Therefore
\[
S_{n,\lambda}^Y(x)=\mathcal{E}_{n,\lambda}(x).
\]
In this case the probabilistically defined Sheffer family coincides exactly with the degenerate Euler polynomials.

The independent-sum formalism extends to \(Y^{(m)}=Y_1+\cdots+Y_m\), where the \(Y_i\) are i.i.d. copies. For \(\mathrm{Bernoulli}(1/2)\), one obtains
\[
S_{n,\lambda}^{Y^{(m)}}(x)=\mathcal{E}_{n,\lambda}^{(m)}(x),
\]
the \(m\)th-order Euler polynomials. For \(\mathrm{Uniform}[0,1]\), the closed form of the previous theorem extends via the factor \((\log(1+\lambda t)/(\lambda t))^m\).

The same framework also produces new addition and connection formulas among higher-order degenerate Bernoulli and Euler polynomials, including
\[
\beta_{n,\lambda}^{(a)}(x+y)
=
\sum_{k=0}^n
\binom{n}{k}
\left[
\beta_{k,\lambda}^{(a)}(x)+\frac{k}{2}\beta_{k-1,\lambda}^{(a-1)}(x)
\right]
\mathcal{E}_{n-k,\lambda}(y),
\]
and
\[
\mathcal{E}_{n,\lambda}^{(b)}(x+y)
=
\sum_{k=0}^n
\binom{n}{k}
\left[
\frac{2}{k+1}\beta_{n-k,\lambda}(x)
\right]
\left[
\mathcal{E}_{k+1,\lambda}^{(b-1)}(y)-\mathcal{E}_{k+1,\lambda}^{(b)}(y)
\right].
\]
The cited exposition characterizes these identities as genuinely new and as arising only in the degenerate hybrid-Sheffer framework [2507.20167].

## 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
\[
D_\lambda p(x)=\frac{p(x+\lambda)-p(x)}{\lambda},
\]
and a sequence \(\{S_n(x)\}_{n\ge 0}\) is called a \(\lambda\)-Sheffer sequence for \((g(t),f(t))\) if it satisfies the corresponding operator conditions, equivalently possessing an exponential generating function of the form
\[
g(t)^{-1}e_\lambda\!\bigl(xf(t)\bigr)
=
\sum_{n=0}^\infty S_n(x)\frac{t^n}{n!}.
\]
Within this framework, fully degenerate Bell and fully degenerate Dowling polynomials are treated as \(\lambda\)-Sheffer sequences, and their expansions, addition theorems, and recurrences are developed through \(\lambda\)-umbral calculus [2108.11090]. Related work on degenerate Hermite polynomials likewise uses the general \(\lambda\)-Sheffer connection formula to represent higher-order degenerate Bernoulli, Euler, and Frobenius–Euler polynomials in a Hermite basis and conversely [2010.14696].

This broader context clarifies an important terminological point. In [2507.20167], “degenerate Sheffer-type polynomials” denotes the specific hybrid family
\[
\left(\frac{t}{e_\lambda(t)-1}\right)^a
\left(\frac{2}{e_\lambda(t)+1}\right)^b
e_\lambda^x(t),
\]
whereas “degenerate Sheffer polynomials associated with \(Y\)” denotes the probabilistic family
\[
\bigl(E[e_\lambda^Y(t)]\bigr)^{-1}e_\lambda^x(t).
\]
In related λ-umbral-calculus papers, by contrast, “degenerate Sheffer” or “\(\lambda\)-Sheffer” denotes the general class attached to a pair \((g,f)\). This suggests that the 2025 construction is best understood not as an isolated nomenclature, but as a specialized realization of the larger \(\lambda\)-Sheffer paradigm, with Bernoulli–Euler hybridization and random-variable attachment as its distinguishing features.

Source: https://www.emergentmind.com/topics/degenerate-sheffer-polynomials