[Paper Review] Series Representation of the Modified Bessel Functions

K. Maślanka|ArXiv.org|Apr 12, 2001

Fractional Differential Equations Solutions2 references3 citations

TL;DR

This paper derives novel power series representations for modified Bessel functions $ K_\alpha $ using fractional derivatives, particularly the Riemann-Liouville formalism. By expressing $ K_\alpha $ as fractional derivatives of exponential and power functions, the author constructs two new hypergeometric-like series expansions—one based on $ V_k^{(-1)} $ polynomials and another on $ V_k^{(-1/2)} $, both converging to $ K_\alpha(z) $ with Pochhammer symbol ratios resembling the Kummer confluent hypergeometric function.

ABSTRACT

Some power series representations of the modified Bessel functions (McDonald functions $K_α$) are derived using the relatively little known formalism of fractional derivatives. The resulting summation formulae are believed to be new.

Motivation & Objective

• To derive new power series representations for the modified Bessel functions $ K_\alpha $, particularly for non-integer orders.
• To explore the utility of fractional calculus—specifically Riemann-Liouville fractional derivatives—in generating series expansions for special functions.
• To define and analyze the polynomial families $ V_k^{(\alpha)}(z) $ as tools for expressing fractional derivatives of exponential-type functions.
• To establish connections between definite integrals involving $ \exp(-\beta/t) $ and the modified Bessel functions through fractional derivative interpretations.
• To produce hypergeometric-like series expansions for $ K_\alpha(z) $ with explicit coefficients derived from generalized binomial and Pochhammer symbols.

Proposed method

• Utilizes the Riemann-Liouville fractional derivative definition for $ s < 0 $, extended to $ s \geq 0 $ via composition with integer-order derivatives.
• Applies the Leibniz rule for fractional derivatives of products, using generalized binomial coefficients $ \binom{s}{j} $ defined via the Gamma function.
• Introduces the polynomial $ V_k^{(\alpha)}(z) = x^k \exp(\beta x^\alpha) \partial_x^k \exp(-\beta x^\alpha) $, proven to be a polynomial in $ z = \beta x^\alpha $ with coefficients $ A_{kj}^{(\alpha)} $ involving ratios of Gamma functions.
• Derives the fractional derivative of $ x^\nu \exp(-\beta x^\alpha) $ using the Leibniz rule and the power rule for fractional derivatives, resulting in expression (9).
• Substituting the fractional derivative expressions into known integral representations (4) and (5) yields two new series expansions for $ K_s(z) $, labeled (10) and (11).
• Transforms the resulting series into hypergeometric-like forms by expressing coefficients in terms of Pochhammer symbols $ (a)_k $, resembling the confluent hypergeometric function $ {}_1F_1 $.

Experimental results

Research questions

• RQ1 Can fractional derivatives be used to derive new series expansions for the modified Bessel function $ K_\alpha(z) $?
• RQ2 What is the structure of the polynomial $ V_k^{(\alpha)}(z) $, and how does it arise from the fractional derivative of $ \exp(-\beta x^\alpha) $?
• RQ3 How do the integral representations (4) and (5) relate to fractional derivatives of $ \exp(-\beta/x) $ and $ \exp(-\beta/\sqrt{x}) $?
• RQ4 Can the resulting series for $ K_s(z) $ be expressed in a hypergeometric-like form with Pochhammer symbol ratios?
• RQ5 What are the convergence properties and explicit coefficient structures of the derived series expansions for $ K_\alpha(z) $?

Key findings

• The modified Bessel function $ K_s(z) $ is expressed as a hypergeometric-like series: $ K_s(z) = 2^{s-1}\Gamma(s)z^{-s}\exp(-z)\left[1 + \sum_{k=1}^{\infty} \frac{(\frac{1}{2}-s)_k}{(\frac{1}{2}+s)_k} \sum_{j=1}^{k} \binom{k-1}{j-1} \frac{(-2z)^j}{j!} \right] $, derived from the $ V_k^{(-1)} $ polynomial family.
• A second series expansion is derived using $ V_k^{(-1/2)}(z) $, yielding $ K_s(z) = 2^{s-1}\sqrt{\pi} \frac{\Gamma(2s)}{\Gamma(\frac{1}{2}-s)} z^{-s} \exp(-z) \sum_{k=0}^{\infty} \frac{(-1)^k}{k!} \frac{\Gamma(k + \frac{1}{2} - s)}{\Gamma(k + \frac{1}{2} + s)} V_k^{(-1/2)}(z) $, valid for $ \text{Re}(s) > -\frac{1}{2} $.
• The coefficients of the $ V_k^{(-1)} $ polynomials are explicitly given by $ A_{kj}^{(-1)} = \frac{(-1)^{k+j}}{(k-j)!} \frac{k!(k-1)!}{j!(j-1)!} $, confirming their polynomial nature and enabling exact series computation.
• The fractional derivative of $ x^\nu \exp(-\beta x^\alpha) $ is derived as $ x^{\nu - s} \frac{\Gamma(\nu+1)}{\Gamma(-s)} \exp(-\beta x^\alpha) \sum_{k=0}^{\infty} \frac{(-1)^k}{k!} \frac{\Gamma(k - s)}{\Gamma(k - s + \nu + 1)} V_k^{(\alpha)} $, forming the core of the method.
• The integral representations (4) and (5) are reinterpreted as fractional derivatives: $ \partial_x^s [x^{2s} \exp(-\beta/x)] = \frac{\beta^{s+1/2}}{\sqrt{\pi x}} \exp(-\beta/(2x)) K_{s+1/2}(\beta/(2x)) $, linking them directly to $ K_\nu $.
• The resulting series expansions are termed 'hypergeometric-like' due to the presence of ratios of Pochhammer symbols $ \frac{(\frac{1}{2}-s)_k}{(\frac{1}{2}+s)_k} $, which mirror the structure of the confluent hypergeometric function $ {}_1F_1 $.