Powers of modified Bessel functions of the first kind

For ν an unrestricted real (or complex) number, let Iν be the modified Bessel function of the first kind of order ν, defined by [1, p. 77] Iν(x) = ∑ n≥0 (x/2)2n+ν n!0(ν + n+ 1) , which occurs frequently in problems of electrical engineering, finite elasticity, quantum billiards, wave mechanics, mathematical physics and chemistry, etc. Here x is an arbitrary real (or complex) number, and as usual, 0 denotes the Euler gamma function. As the authors of [2] remarked, the products of Bessel and of modified Bessel functions of the first kind appear frequently in problems of statistical mechanics and plasma physics [3]. Motivated by the importance of products of modified Bessel functions in [2] the authors deduced explicit representations for powers of these functions. In this note we would like to point out that although little is known about the explicit expression for the MacLaurin series of powers of modified Bessel functions of the first kind, the general theory is very old and is well developed. In [2, Eq. 30] the authors deduced an explicit recurrence formula which determines the coefficients in the MacLaurin series expansion of powers of modified Bessel functions of the first kind, i.e. they proved that [Iν(x)] = ∑ n≥0 an(r)(x/2) n!(ν + n)!(ν!)r−1 , (1) where the polynomials an(r) are determined recursively by [2, Eq. 31] an(r) = r ν + n ν + 1 an−1(r)+ n ∑ k=0 bk(ν) (ν + 1)!Ck ν+n n!(ν + k+ 1)! an−1(r), (2) and the integer sequence bk(ν) is identified by expanding the right-hand side of ∑ k≥0 bk(ν)x (ν + k)!(k− 1)! = x (ν + 1)! [ √ xIν(2 √ x) (ν + 1)Iν+1(2 √ x) − 2 ] . I Research supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. E-mail address: [email protected]. 0893-9659/$ – see front matter© 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2010.02.015 Á. Baricz / Applied Mathematics Letters 23 (2010) 722–724 723 Here, as usual, Ck n = ( n k ) = n! k!(n− k)! denotes the binomial coefficient. Although in [2] the ranges of validity of ν and r are not specified, due to the notation tacitly it is assumed that ν is a natural number and r is an arbitrary real number. In this short note we show that in fact (1) can be deduced easily for ν and r real (or complex) numbers by using a very old formula of Euler; moreover instead of (2) we propose another recurrence formula for the coefficients which is more convenient for direct computations. To this end let r be an arbitrary real (or complex) number and consider the power series f (x) = ∑ n≥0 cnx n and [f (x)]r = ∑ n≥0 dnx n. Thus [4, p. 754] d0 = cr 0, c0 6= 0, dn = 1 nc0 n ∑ k=1 [k(r + 1)− n]ckdn−k. (3) This recurrence formula was deduced first in 1748 by the genius of the ‘‘teacher of all mathematicians’’ whose iconic name is Leonhard Euler, in his famous Introductio in Analysin Infinitorum. However, this basic recurrence formula is not as widely known as it should be, and it has been rediscovered several times. For more details on the history of relation (3) and related coefficient problems in multiplying power series the interested reader is referred to [5] and to the references therein. Now consider the power series Iν(x) = 20(ν + 1)xIν(x) = ∑ n≥0 (x/4)n n!(ν + 1)n , which is sometimes called the normalized modified Bessel function of the first kind. Here (ν + 1)n for ν 6= −1,−2, . . . denotes the well-known Pochhammer (or Appell) symbol defined in terms of Euler’s gamma function, i.e. (ν + 1)n = (ν + 1)(ν + 2) · · · (ν + n) = 0(ν + n + 1)/0(ν + 1). For convenience we denote by αn the general coefficient of the above power series, i.e. let αn = [4nn!(ν + 1)n] for all n ≥ 0 integer. Moreover we assume that the power series of [Iν(x)] has the form ∑ n≥0 βn(r)x n. Observe that α0 = 1 and hence β0(r) = 1. Then by using Euler’s formula (3) we easily get