[Paper Review] Some identities of polynomials arising from umbral calculus

Dae San Kim, Taekyun Kim+1 more|arXiv (Cornell University)|Nov 15, 2012

Advanced Mathematical Identities5 references3 citations

TL;DR

This paper derives new identities for special polynomials—such as Bernoulli, Euler, and generalized Appell-type polynomials—using umbral calculus techniques. By leveraging associated sequences and generating functions, the authors establish explicit formulas involving Stirling numbers, Bernoulli numbers of higher order, and binomial coefficients, yielding closed-form expressions for polynomial sequences via operational calculus on formal power series.

ABSTRACT

In this paper, we study some properties of associated sequaences in umbral calculus. From these properties, we derive new and interesting identities of several kinds of polynomials.

Motivation & Objective

• To investigate structural properties of associated sequences in umbral calculus for deriving new polynomial identities.
• To establish connections between polynomial sequences and formal power series through linear functionals and generating functions.
• To derive explicit formulas for generalized Bernoulli and Euler polynomials using operational methods in umbral algebra.
• To unify and extend known identities by expressing polynomials in terms of Stirling numbers and binomial coefficients.
• To provide closed-form expressions for sequences such as $ A_n(x;a) $, $ S_n(x) $, and $ E_n^{(n)}(x) $ via umbral operations.

Proposed method

• Utilizes the umbral algebra framework, treating formal power series as linear functionals on polynomials.
• Applies the operational identity $ x o x rac{f(t)}{g(t)} x^{-1} $ to relate associated sequences $ p_n(x) \sim (1,f(t)) $ and $ q_n(x) \sim (1,g(t)) $.
• Employs generating functions for Bernoulli $ \left(\frac{t}{e^t-1}\right)^r $, Euler $ \left(\frac{2}{e^t+1}\right)^r $, and falling factorial polynomials.
• Expresses $ \left(\frac{e^t-1}{t}\right)^n $ as a series in $ t^l $ involving Stirling numbers of the second kind $ S_2(l+n,n) $.
• Uses the relation $ (x)_{n-1} = \sum_{l=0}^{n-1} S_1(n-1,l) x^l $ to expand polynomials in monomial bases.
• Applies the operational calculus identity $ t^k p(x) = \frac{d^k}{dx^k} p(x) $ and $ e^{yt} p(x) = p(x+y) $ to manipulate polynomial sequences.

Experimental results

Research questions

• RQ1 How can associated sequences in umbral calculus be used to derive new identities for generalized Bernoulli and Euler polynomials?
• RQ2 What closed-form expressions emerge when applying umbral operations to sequences like $ x(x-an)^{n-1} $ and $ E_n^{(n)}(x) $?
• RQ3 How do Stirling numbers of the first and second kind appear in the expansion of polynomial sequences under umbral operations?
• RQ4 What is the operational relationship between $ S_n(x) \sim (1, t(1+t)^a) $ and monomials $ x^n $, and how can it be expressed algebraically?
• RQ5 Can higher-order Bernoulli numbers $ B_k^{(k-n+1)}(an+1) $ be linked to polynomial sequences via generating functions in the umbral framework?

Key findings

• The paper derives a closed-form expression for $ A_n(x;a) = x(x - an)^{n-1} $ as a sum over falling factorials involving Stirling numbers and binomial coefficients.
• It proves that $ S_n(x) \sim (1, t(1+t)^a) $ satisfies $ S_n(x) = \sum_{k=1}^n \binom{-an}{n-k} (n-1)_{n-k} x^k $, linking it to generalized binomial coefficients.
• The identity $ xE_{n-1}^{(n)}(x) = \sum_{k=0}^{n-1} \binom{n-1}{k} E_k^{(n)}(an) A_{n-k}(x - ak; a) $ is established, connecting Euler polynomials to generalized Appell polynomials.
• A new identity is found: $ S_2(n,m) = \sum_{l=m}^n \binom{-an}{n-l} (n-1)_{n-l} \binom{l-1}{m-1} B_{l-m}^{(l-m-n+1)}(an+1) $, relating Stirling numbers to higher-order Bernoulli numbers.
• The generating function $ \left(\frac{t(1+t)^a}{\log(1+t)}\right)^n = \sum_{k=0}^\infty B_k^{(k-n+1)}(an+1) \frac{t^k}{k!} $ is derived and used to express polynomial sequences.
• By setting $ b=1 $, the identity for $ A_n(x;a) $ simplifies to a sum involving $ (-an)^{k-j} $, $ S_2(j+n,n) $, $ S_1(n-1,l) $, and binomial coefficients.