[Paper Review] A probabilistic interpretation of the Volkenborn integral

Ashwani K. Bhandari, Christophe Vignat|arXiv (Cornell University)|Jan 18, 2012

Advanced Mathematical Identities3 references3 citations

TL;DR

This paper provides a probabilistic interpretation of the Volkenborn integral by linking it to expectations involving logistic and hyperbolic secant distributed random variables, enabling new derivations of identities for Bernoulli and Euler polynomials. The key contribution is a multivariate extension of Raabe’s multiplication theorem using these probabilistic representations.

ABSTRACT

In this paper, we provide a probabilistic interpretation of the Volkenborn integral; this allows us to extend results by T. Kim et al about sums of Euler numbers to sums of Bernoulli numbers. We also obtain a probabilistic representation of the multidimensional Volkenborn integral which allows us to derive a multivariate version of Raabe's multiplication theorem for the higher-order Bernoulli and Euler polynomials.

Motivation & Objective

• To establish a probabilistic framework for the Volkenborn integral using special continuous and discrete random variables.
• To extend known identities for Euler numbers to Bernoulli numbers via this probabilistic interpretation.
• To derive a multivariate version of Raabe’s multiplication theorem for higher-order Bernoulli and Euler polynomials.
• To handle both even and odd cases of the multiplication parameter m in Raabe’s identity using signed measures and cancellation properties.
• To unify and generalize results from Kim et al. on Bernstein polynomials to Beta polynomials in the context of Volkenborn integrals.

Proposed method

• Represent the q=0 Volkenborn integral as the expectation of f(x + iLB - 1/2), where LB follows the logistic distribution with density (π/2)sech²(πx).
• Represent the q=1 Volkenborn integral as the expectation of f(x + iLE - 1/2), where LE follows the hyperbolic secant distribution with density sech(πx).
• Use the cancellation property: if UB ~ Uniform[0,1] independent of LB, then E[(x + iLB - 1/2 + UB)^n] = x^n.
• Apply the same principle for Rademacher variables UE: E[(x + iLE - 1/2 + UE)^n] = x^n.
• Derive multivariate identities by introducing independent copies of the random variables and using signed measures for the Euler case.
• Leverage the fact that sums of independent uniform and Rademacher variables can be reweighted to match scaled versions of the original variables, enabling cancellation of stochastic terms.

Experimental results

Research questions

• RQ1 How can the Volkenborn integral be interpreted probabilistically using known probability distributions?
• RQ2 Can the probabilistic representation of the Volkenborn integral be used to derive new identities for Bernoulli numbers and polynomials?
• RQ3 What is the multivariate generalization of Raabe’s multiplication theorem for Bernoulli and Euler polynomials?
• RQ4 How do the results differ when the multiplication parameter m is even versus odd?
• RQ5 Can the cancellation properties of uniform and Rademacher variables be used to simplify multivariate Volkenborn integrals?

Key findings

• The q=0 Volkenborn integral of f(x) equals E[f(x + iLB - 1/2)], where LB has the logistic distribution with density (π/2)sech²(πx).
• The q=1 Volkenborn integral of f(x) equals E[f(x + iLE - 1/2)], where LE has the hyperbolic secant distribution with density sech(πx).
• The n-th Bernoulli number satisfies B_n = E[(iLB - 1/2)^n], and the n-th Euler number satisfies E_n = E[(iLE - 1/2)^n].
• For the multivariate case, Raabe’s identity for Bernoulli polynomials is derived as m^{-n} E[(mx + m∑a_i(iLB^{(i)} - 1/2) + ∑a_i(iL̃B^{(i)} - 1/2) + m∑a_i U_B^{(i)})^n] = B_n^{(k)}(mx|a).
• For Euler polynomials with odd m, the identity ∑_{l_i=0}^{m-1} (-1)^{∑l_i} E_n(x + (1/m)∑a_i l_i | a) = m^{-n} E[(mx + m∑a_i(iLE^{(i)} - 1/2) + ∑a_i(iL̃E^{(i)} - 1/2) + m∑a_i U_E^{(i)})^n] holds.
• For even m, a signed measure is introduced over {0,…,m-1} with weights (−1)^l, and the identity involves a difference of expectations that reduces to a derivative-like term, yielding E_n^{(k)}(mx) up to a factor.