[Paper Review] Complete Bell polynomials and new generalized identities for polynomials of higher order

Boris Rubinstein|ArXiv.org|Nov 16, 2009

Advanced Mathematical Identities6 references3 citations

TL;DR

This paper establishes new identities for higher-order Bernoulli and Eulerian polynomials by expressing them as complete Bell polynomials, revealing deep algebraic connections. The key contribution is a general identity linking these special polynomials to Bell polynomials via power sums and Bernoulli numbers, leading to novel recurrence relations and generalized Euler identities for higher-order cases.

ABSTRACT

The relations between the Bernoulli and Eulerian polynomials of higher order and the complete Bell polynomials are found that lead to new identities for the Bernoulli and Eulerian polynomials and numbers of higher order. General form of these identities is considered and generating function for polynomials satisfying this general identity is found.

Motivation & Objective

• To establish a systematic connection between higher-order Bernoulli and Eulerian polynomials and complete Bell polynomials.
• To derive new recurrence identities for higher-order Bernoulli and Eulerian polynomials using the structure of complete Bell polynomials.
• To generalize known Euler and Bernoulli identities to higher-order cases through the use of parameterized generating functions.
• To provide a unified framework for generating functions of polynomials satisfying a generalized Bell-type identity.
• To recover and extend classical identities, including Nörlund’s identity, within a broader algebraic context.

Proposed method

• Expressing the generating functions of higher-order Bernoulli and Eulerian polynomials in terms of exponential generating functions involving power sums and Bernoulli numbers.
• Using the Taylor expansion of logarithmic generating functions to relate the coefficients to complete Bell polynomials via cumulants and moments.
• Deriving a general identity where higher-order polynomials are represented as complete Bell polynomials with coefficients involving Bernoulli numbers and power sums.
• Applying the generating function of complete Bell polynomials to derive recurrence relations for higher-order polynomials.
• Transforming the general identity into explicit recurrence relations by substituting power sum expressions and simplifying using binomial coefficients.
• Deriving new identities by combining the Bell polynomial representation with known identities for Bernoulli numbers and polynomials, including the Euler identity.

Experimental results

Research questions

• RQ1 How can higher-order Bernoulli and Eulerian polynomials be systematically related to complete Bell polynomials?
• RQ2 What generalized recurrence identities emerge from expressing these polynomials as complete Bell polynomials?
• RQ3 Can the classical Euler identity for Bernoulli numbers be extended to higher-order Bernoulli polynomials using this framework?
• RQ4 What is the generating function for polynomials satisfying the generalized Bell-type identity derived in this work?
• RQ5 How do the new identities recover or generalize known results, such as Nörlund’s identity for higher-order Bernoulli polynomials?

Key findings

• The higher-order Bernoulli polynomial $ B^{(m)}_n(s|oldsymbol{d}) $ is exactly equal to the complete Bell polynomial $ oldsymbol{B}_n(s + a_1, a_2, ar{a}) $, where $ a_i = (-1)^{i-1} B_i au_i / i $ and $ au_k = ext{tr}(oldsymbol{d}^k) $.
• A generalized identity is derived: $ B^{(m)}_n(s|oldsymbol{d}) = (s - au_1) B^{(m)}_{n-1}(s|oldsymbol{d}) - rac{1}{n} inom{n}{k} B_k au_k B^{(m)}_{n-k}(s|oldsymbol{d}) $, which reduces to known identities upon specialization.
• The identity $ au_1 B^{(m)}_n(s|oldsymbol{d}) = (s - au_1) B^{(m)}_{n-1}(s|oldsymbol{d}) - rac{1}{n} au_k B_k B^{(m)}_{n-k}(s|oldsymbol{d}) $ leads to a new recurrence for higher-order Bernoulli polynomials.
• A new identity is established: $ au_1 B^{(m)}_n(s|oldsymbol{d}) = (s - au_1) B^{(m)}_{n-1}(s|oldsymbol{d}) - rac{1}{n} au_k B_k B^{(m)}_{n-k}(s|oldsymbol{d}) $, which generalizes the classical Euler identity.
• For $ oldsymbol{d} = oldsymbol{1} $, the identity reduces to Nörlund’s known result: $ m B^{(m+1)}_n(s|oldsymbol{1}) = (m - n) B^{(m)}_n(s|oldsymbol{1}) + n(s - m) B^{(m)}_{n-1}(s|oldsymbol{1}) $.
• For $ m = n $, the identity yields $ B^{(m+1)}_m(s|oldsymbol{1}) = (s - m) B^{(m)}_{m-1}(s|oldsymbol{1}) = au_{m-1} $, confirming a known product identity for Bernoulli polynomials at $ m = n $.