Higher Derivatives of Airy Functions and of their Products

Abstract

The problem of evaluation of higher derivatives of Airy functions in a closed form is investigated. General expressions for the polynomials which have arisen in explicit formulae for these derivatives are given in terms of particular values of Gegenbauer polynomials. Similar problem for products of Airy functions is solved in terms of terminating hypergeometric series.

Asymptotic solutions of inhomogeneous differential equations having a turning point

“…This path connects z (1) to z (0) , passes through z, and meets the monotonicity requirement that ℜ(z 3/2 ) be nondecreasing as t passes along it from z (1) to z (0) . Moreover, the inclusion of the arc segment ensures that |t| ≥ |z| for all t on the path, (0) and hence the bounds (6.23) and (6.24) are of the appropriate order of magnitude as z → ∞.…”

“…This lemma can be used to give a general representation of the iterated integration of the Airy functions, but we do not give details since they are not applicable here. We mention that in [1] similar expressions were derived for the higher derivatives of Airy functions in a closed form, which also involve the Airy function, its first derivative and certain polynomials.…”

“…for some constants c m,0 (u) and c m,1 (u), and where w m,j (u, z) (j = 0, 1) are the homogeneous solutions given by (2.15). On letting z → z (1) we see that all functions in (5.1) are bounded, with the exception of w m,0 (u, z), which implies c m,0 (u) = 0.…”

“…present the main results from [9] and [10]. Firstly, here and elsewhere we that assume f (z) and g(z), if meromorphic in Z, have poles at finite points z = w j (j = 1, 2, 3, • • • ), say, such that at z = w j (i) f (z) has a pole of order m > 2, and g(z) is analytic or has a pole of order less than 1 2 m + 1, or (ii) f (z) and g(z) have a double pole, and (z − w j ) 2 g(z) → − 1 4 as z → w j . We shall call these admissible poles.…”

Asymptotic solutions of inhomogeneous differential equations having a turning point

“…We mention that in Ref. 17 similar expressions were derived for the higher derivatives of Airy functions in a closed form, which also involve the Airy function, its first derivative and certain polynomials.…”