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Mathematical Postulations
Published in Seshu Kumar Damarla, Madhusree Kundu, Fractional Order Processes, 2018
Seshu Kumar Damarla, Madhusree Kundu
For the special case of p=2 and q=1, the generalized hypergeometric function in Equation (1.14) produces the well-known Gauss hypergeometric function
Series Solutions of Second Order ODEs
Published in K.T. Chau, Theory of Differential Equations in Engineering and Mechanics, 2017
There are also a number of important functions that can be expressed in terms of the generalized hypergeometric function 1F1(n+1/2,2n+1,2ix)=(2x)nn!eixJn(x) $$ _{1} F_{1} (n + 1/2,2n + 1,2ix) = (\frac{2}{x})^{n} n!e^{{ix}} J_{n} (x) $$ 1F1(-n,1/2,x2)=(-1)nn!(2n)!H2n(x) $$ _{1} F_{1} ( - n,1/2,x^{2} ) = ( - 1)^{n} \frac{{n!}}{{(2n)!}}H_{{2n}} (x) $$ 1F1(-n,3/2,x2)=(-1)nn!2(2n+1)!H2n+1(x) $$ _{1} F_{1} ( - n,3/2,x^{2} ) = ( - 1)^{n} \frac{{n!}}{{2(2n + 1)!}}H_{{2n + 1}} (x) $$
Analysis of the family of integral equation involving incomplete types of I and Ī-functions
Published in Applied Mathematics in Science and Engineering, 2023
Sanjay Bhatter, Kamlesh Jangid, Shyamsunder Kumawat, Dumitru Baleanu, D. L. Suthar, Sunil Dutt Purohit
If we substitute in Theorem 3.3, then we obtain the Fredholm integral equation solution that uses the incomplete generalized hypergeometric function . Solution: Suppose denotes the component of Equation (43) of Example 5.4. Thereupon, using Lemma 3.1 and the definition in Equation (20), we obtain Next, by altering the order of integration under the allowable circumstances, we obtain Moreover, using the widely used definition of the RL fractional derivative, we get Now, replacing , then we get the final solution of Equation (43).
Regimes of optical propagation through turbulence: theory and direct numerical simulations
Published in Waves in Random and Complex Media, 2023
Another simplifying assumption made in classical analyses is that the outer scale () or the largest scales of refractive index fluctuations are infinitely large, that is, . The effect of finite outer scale can be incorporated in the limits of the integral in Equation (9) as At very short distances (formally ), substituting Equation (12) in Equation (15) yields where is a Kummer confluent hypergeometric function and is a generalized hypergeometric function [33, 34]. Equation (16) reduces to Equation (14) when implying that the effects of outer-scale are weak at such short distances and relevant only at low Reynolds number. Similar expressions can also be derived for the and regimes. This is done next.
Fourier multipliers for nonlocal Laplace operators
Published in Applicable Analysis, 2021
In this section, we derive formulas for the Fourier multipliers of utilizing the generalized hypergeometric function. To begin, we express through its Fourier transform as Since the definition of can be extended to the space of tempered distributions through the multipliers derived below, it is sufficient to assume that u is a Schwartz function. Applying then shows that providing the representation are the Fourier multipliers of the operator . Since the integrand in (7) is integrable as long as . The imaginary part of (7) vanishes due to symmetry, so an alternative form is Note that the second integral in (8) is finite for . Thus we may consider the formula (8) as the definition for with .