Uniform Asymptotic Expansions of Laguerre Polynomials

Asymptotic Approximations between the Hahn-Type Polynomials and Hermite, Laguerre and Charlier Polynomials

“…3.1 to obtain(2).The function φ(w) = log f (w) verifies Lemma 1 with μ = c, s = 1 and m = 3. Therefore, we have c k = O(c[k/3] ).…”

“…Substituting the Maclaurin series of the logarithm of the above 3 F 2 function into its differential equation [10], and following a similar argument as in Sect. 4.5 we find that the Taylor coefficients at w = 0 of the logarithm of the above 3 Substitute: 2 and On the other hand, A = O(a), X = O(a) and, taking into account that lim a→∞ A/X = 1, we have C n−k (X, A) = O(a 0 ) and we obtain the asymptotic behaviour 2.9.2. The limit (31) follows from the first term of (28) after obtaining x 2 (X, A) and c(X, A).…”

“…On the other hand, our method gives asymptotic expansions of polynomials situated at any level of the table in terms of polynomials located at lower levels. Our method is also different from the sophisticated uniform methods considered for example in [9] or [2], where asymptotic expansions of the Meixner M n (nx, b, c) or Charlier C a n (nx) polynomials respectively are given for large values of n and fixed a, b, c, x. In our method we keep the degree n fixed and let some parameter(s) of the polynomial go to infinity.…”

Heat kernel asymptotic expansions for the Heisenberg sub-Laplacian and the Grushin operator

“…Note that θ 1 < π < θ 2 for x = 0, and that θ 1 and θ 2 both tend to π as x → 0, and τ = iπ is also a simple pole of the phase function f (τ ). For such a case, Frenzen & Wong [28] derived a uniform asymptotic approximation in terms of Bessel function; see also [22,ch. VII].…”

“…Follow the approach of Frenzen & Wong [28], we can derive an asymptotic expansion of the form 6) as t → 0 + .…”