Asymptotic expansions for Riesz fractional derivatives of Airy functions and applications

Riesz fractional derivatives of a function, (also called Riesz potentials), are defined as fractional powers of the Laplacian. Asymptotic expansions for large x are computed for the Riesz fractional derivatives of the Airy function of the first kind, Ai(x), and the Scorer function, Gi(x). Reduction formulas are provided that allow one to express Riesz potentials of products of Airy functions, and , via and . Here Bi(x) is the Airy function of the second type. Integral representations are presented for the function A₂(a,b;x)=Ai(x−a)Ai(x−b) with a,b∈R and its Hilbert transform. Combined with the above asymptotic expansions they can be used for computing asymptotics of the Hankel transform of . These results are used for obtaining the weak rotation approximation for the Ostrovsky equation (asymptotics of the fundamental solution of the linearized Cauchy problem as the rotation parameter tends to zero).     

A Rearrangement of 3‐Pyrazolines as a Missing Link

The thermal conversion of 4‐isoxazolines to 4‐oxazolines involves the transposition of two ring members. The ring‐contraction and ring‐expansion sequence in the reaction 2 → 5 has been previously clarified. The low N−N bond energy should favor an analogous conversion of 3‐pyrazolines 6 to 4‐imidazolines 7 ; the first example of such a transformation is reported here. In the yellow 16 , the 3‐pyrazoline is part of a pyrazolo[5,1‐a]isoquinoline system. Daylight induces a ring contraction, which affords the 2‐isoquinolylaziridine derivative 21 . The latter is converted at 65° to the tricyclic 4‐imidazoline 26 by a sequence of electrocyclic aziridine ring‐opening and 1,5‐electrocyclization of a C=N‐conjugated azomethine ylide 25 .     

On the expansion of confluent hypergeometric functions in terms of Bessel functions

For the confluent hypergeometric functions U (a, b, z) and M (a, b, z) asymptotic expansions are given for a → ∞. The expansions contain modified Bessel functions. For real values of the parameters rigorous error bounds are given.     

Asymptotics and numerics of polynomials used in Tricomi and Buchholz expansions of Kummer functions

Expansions in terms of Bessel functions are considered of the Kummer function ₁ F ₁(a; c, z) (or confluent hypergeometric function) as given by Tricomi and Buchholz. The coefficients of these expansions are polynomials in the parameters of the Kummer function and the asymptotic behavior of these polynomials for large degree is given. Tables are given to show the rate of approximation of the asymptotic estimates. The numerical performance of the expansions is discussed together with the numerical stability of recurrence relations to compute the polynomials. The asymptotic character of the expansions is explained for large values of the parameter a of the Kummer function.