[Paper Review] On p-adic analogue of q-Bernstein polynomials and related integrals

Taekyun Kim, Lee-Chae Jang+2 more|arXiv (Cornell University)|Sep 17, 2010

Advanced Mathematical Identities9 references4 citations

TL;DR

This paper introduces a p-adic analogue of Kim's q-Bernstein polynomials and derives explicit formulas for the p-adic q-integral of products of these polynomials and their powers. By leveraging Carlitz's q-Bernoulli numbers and polynomials, the authors establish new identities connecting p-adic q-integrals to generalized binomial sums and q-integer expressions, providing a p-adic framework for q-Bernstein polynomial theory in number theory and special functions.

ABSTRACT

In this paper, we give p-adic q-integral representation for the Kim's q-Bernstein polynomials and we give some interesting formulae realted to Carlitz's q-Bernoulli numbers.

Motivation & Objective

• To develop a p-adic analogue of Kim's q-Bernstein polynomials for use in p-adic analysis.
• To express the p-adic q-integral of these polynomials using Carlitz's q-Bernoulli numbers and polynomials.
• To derive closed-form identities for the p-adic q-integral of products of multiple q-Bernstein polynomials and their powers.
• To establish connections between p-adic q-integrals and generalized binomial sums involving q-integers.

Proposed method

• Define Kim's q-Bernstein polynomials using q-integers [x]_q and [1-x]_{1/q}.
• Utilize the p-adic q-integral on ℤ_p via the limit of Riemann-type sums with q-measures.
• Apply the representation β_{m,q} = ∫_{ℤ_p} [x]_q^m dμ_q(x) to link integrals to Carlitz’s q-Bernoulli numbers.
• Use binomial expansions and generating function techniques to decompose products of Bernstein polynomials into monomial integrals.
• Apply symmetry and duality relations between q and 1/q to derive equivalent integral expressions.
• Derive closed-form identities by expressing integrals of monomials [x]_q^k as β_{k,q} and using inclusion-exclusion via binomial coefficients.

Experimental results

Research questions

• RQ1 How can Kim’s q-Bernstein polynomials be extended to a p-adic setting using p-adic q-integrals?
• RQ2 What is the explicit form of the p-adic q-integral of a single q-Bernstein polynomial on ℤ_p?
• RQ3 How do products of multiple q-Bernstein polynomials integrate p-adically, and what identities emerge?
• RQ4 What role do Carlitz’s q-Bernoulli numbers play in representing these p-adic integrals?
• RQ5 Can symmetric identities be derived between integrals involving q and 1/q parameters?

Key findings

• The p-adic q-integral of the q-Bernstein polynomial B_{k,n}(x,q) is expressed as a finite sum involving binomial coefficients and Carlitz’s q-Bernoulli numbers β_{k,q}.
• A duality identity is established: ∫_{ℤ_p} [1−x + x₁]_{1/q}^n dμ_{1/q}(x₁) = (−1)^n q^n ∫_{ℤ_p} [x + x₁]_q^n dμ_q(x₁).
• For products of s q-Bernstein polynomials of the same index k, the p-adic q-integral is given by a double sum involving β_{sk+l,q} and binomial coefficients.
• When higher powers m_i of the same Bernstein polynomials are considered, the integral becomes a sum over l with terms involving β_{(∑m_i)k + l, q}.
• A symmetric identity is derived: the sum of binomial-weighted terms with β_{n₁+⋯+nₛ−l,1/q} equals the sum of β_{sk+l,q} terms under specific sign and coefficient rules.
• The results generalize classical Bernstein polynomial identities to the p-adic q-setting, providing new tools for p-adic special functions and q-series.