W. Gautschi The incomplete gamma function since Tricomi Tricomiʼs Ideas and Contemporary Applied Mathematics, Atti Conv. Lincei, vol. 147, Accad. Naz. Lincei, Rome (1998), pp. 207-237 Google Scholar [12] F.A. Grünbaum A new type of inequality for Bessel functions J. Math. Anal. ...
The Incomplete Gamma Functions Since Tricomi The incomplete gamma functions since tricomi. Tricomi's Ideas and Contemporary Applied Mathematics, (147):203-237, 1998.W. Gautschi, The incomplete gamma function since Tricomi, in Tricomi's ideas and contemporary applied mathematics, ... W Gautschi ...
For 伪 = 1, the Erlang functions are particular cases of the lower/upper incomplete gamma functions 螕(a, x), corresponding to integer values a = n of the parameter. Note that the lower one equals also a鈭 1xae鈭抶F1,1(1, 1 + a; x) where F1,1(1, 1 + a; x) is the ...
Our main results obtained in Sect.3involve the complementary incomplete gamma function\varGamma (a,z )and the generalized Hurwitz–Lerch zeta function\varPhi _{\mu }^{*} (z,s,a ). Their definitions and various properties are presented in Sect.2. 2Definitions and auxiliary results In this s...
Tricomi FG, Erdélyi A (1951) The asymptotic expansion of a ratio of Gamma functions. Pac J Math 1:133–142 Article MathSciNet MATH Google Scholar Verbeke G, Molenberghs G (2000) Linear mixed models for longitudinal data. Springer, New York MATH Google Scholar Vonesh E, Chinchilli VM...
Equivalently, “output moments” (i.e., Taylor coefficients of functions of interest) are expressed in terms of “input data moments”. In the case of insufficient data, Padé approximations of the output function seem to deserve special attention, because they involve only a few input moments...
To evaluate the inner integral, one can apply the definition of the lower incomplete gamma function (see Equation (8.2.1) in [35]): γ n + 1 , z = ∫ 0 z t n e − t d t = 𝛾(𝑛+1,𝑧)=∫𝑧0𝑡𝑛𝑒−𝑡d𝑡= (10) = ...
To evaluate the inner integral, one can apply the definition of the lower incomplete gamma function (see Equation (8.2.1) in [35]): γ n + 1 , z = ∫ 0 z t n e − t d t = (10) = n ! 1 − e − z ∑ k = 0 n z k k ! , ...
The considered models extend the generalized Black-Scholes-Merton, the variance-gamma and the skew Student’s t-models. The obtained formulas depend on the values of special mathematical functions but can be computed over 0.5 s on modern software. The prices of forward-start put options can be...
In the framework of the theory of linear viscoelasticity, we derive an analytical expression of the relaxation modulus in the Andrade model Gαt for the case of rational parameter α=m/n∈(0,1) in terms of Mittag–Leffler functions from its Laplace trans