[Paper Review] A note on q-Bernoulli numbers and q-Bernstein polynomials

T. Kim, C. S. Ryoo+1 more|arXiv (Cornell University)|Oct 4, 2010

Advanced Mathematical Identities3 references3 citations

TL;DR

This paper introduces extended q-Bernstein polynomials on p-adic integers and investigates their properties via the bosonic p-adic q-integral. It derives explicit formulas for the p-adic q-integral of these polynomials, connecting them to q-Bernoulli numbers and establishing symmetric identities and recurrence relations through generating functions and integral transforms.

ABSTRACT

In this paper we consider the extended q-Bernstein polynomials which are constructed by T. Kim and we investigate some properties.

Motivation & Objective

• To extend Kim’s q-Bernstein polynomials to a two-variable form on p-adic integers for broader analytical application.
• To investigate the bosonic p-adic q-integral of these extended polynomials using uniform differentiability and p-adic measure theory.
• To establish connections between the p-adic q-integral of q-Bernstein polynomials and q-Bernoulli numbers through integral representations.
• To derive symmetric identities and recurrence relations for the extended q-Bernstein polynomials using generating functions and integral identities.
• To generalize results to multiple products of q-Bernstein polynomials and derive closed-form expressions for their p-adic q-integrals.

Proposed method

• Defining extended q-Bernstein polynomials $ B_{k,n}(x_1,x_2 ig| q) = inom{n}{k}[x_1]_q^k[1-x_2]_{q^{-1}}^{n-k} $ for $ x_1,x_2 o [0,1] $, $ 0 < q < 1 $.
• Using the generating function $ F_q^{(k)}(x_1,x_2 ig| t) = rac{(t[x_1]_q)^k ext{exp}(t[1-x_2]_{q^{-1}})}{k!} $ to derive the polynomial coefficients via Taylor expansion.
• Applying the bosonic p-adic q-integral $ I_q(f) = rac{1}{[p^N]_q} o ext{lim}_{N oty} rac{1}{[p^N]_q} ext{sum}_{x=0}^{p^N-1} f(x) q^x $ to evaluate integrals over $ bZ_p $.
• Expressing $ q $-Bernoulli numbers via $ eta_{n,q} = ext{int}_{bZ_p} [x]_q^n dar au_q(x) $, and extending this to polynomial integrals.
• Deriving recurrence relations via partial derivatives: $ rac{ ext{d}}{ ext{d}x_1} B_{k,n} = rac{ ext{log } q}{q-1} n ig( (q-1)[x_1]_q B_{k-1,n-1} + B_{k-1,n-1} ig) $.
• Using binomial expansions and symmetric identities such as $ B_{k,n}(1-x_2,1-x_1 ig| q^{-1}) = B_{k,n}(x_1,x_2 ig| q) $ to simplify multiple integral expressions.

Experimental results

Research questions

• RQ1 How can Kim’s q-Bernstein polynomials be generalized to a two-variable form on $ bZ_p $, and what are their analytic properties?
• RQ2 What is the explicit expression for the bosonic p-adic q-integral of the extended q-Bernstein polynomials?
• RQ3 How do symmetric identities and recurrence relations for the extended polynomials relate to q-Bernoulli numbers?
• RQ4 What is the p-adic q-integral of the product of multiple extended q-Bernstein polynomials, and how does it relate to higher-order q-Bernoulli numbers?
• RQ5 Can generating functions and integral transforms be used to derive closed-form identities for these integrals?

Key findings

• The extended q-Bernstein polynomials satisfy the recurrence $ [1-x_2]_{q^{-1}} B_{k,n}(x_1,x_2 ig| q) + [x_1]_q B_{k-1,n}(x_1,x_2 ig| q) = B_{k,n+1}(x_1,x_2 ig| q) $.
• The p-adic q-integral of $ B_{k,n}(x_1,x_2 ig| q) $ is given by $ inom{n}{k} eta_{k,q} $, linking it directly to q-Bernoulli numbers.
• For $ m,n,k o bZ_+ $, the double integral $ ext{int}_{bZ_p} ext{int}_{bZ_p} B_{k,n} B_{k,m} dar au_q(x_1) dar au_q(x_2) $ yields $ inom{n}{k}inom{m}{k} eta_{2k,q} ext{int}_{bZ_p} [1-x_2]_{q^{-1}}^{n+m-2k} dar au_q(x_2) $.
• A symmetric identity yields $ inom{n}{k} eta_{k,q} = (1 - k - rac{k}{[2]_q}) + ext{sum}_{l=0}^{k-2} inom{k}{l} (-1)^{k+l} (q^2 eta_{k-l,q^{-1}} + k-l+1 - q) $.
• For $ s $-fold products, $ ext{int} ext{prod}_{i=1}^s B_{k,n_i} dar au_q(x_1) dar au_q(x_2) = ext{prod}_{i=1}^s inom{n_i}{k} eta_{sk,q} ext{int}_{bZ_p} [1-x_2]_{q^{-1}}^{n_1+ ext{...}+n_s - sk} dar au_q(x_2) $.
• The general identity for $ s $-fold integrals leads to $ ext{prod}_{i=1}^s inom{n_i}{k} eta_{sk,q} = 1 - sk - rac{sk}{[2]_q} + ext{sum}_{l=0}^{sk-2} inom{sk}{l} (-1)^{sk+l} (q^2 eta_{sk-l,q^{-1}} + sk-l+1 - q) $.