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→The series 1F2: Gave actual connection between Lommel function and generalized hgf, and confirmed the Lommel function definition is found in the actual Watson text. |
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The function <math>x\; {}_1F_2\left(\frac{1}{2};\frac{3}{2},\frac{3}{2};-\frac{x^2}{4}\right)</math> is the antiderivative of the [[cardinal sine]]. With modified values of <math>a_1</math> and <math>b_1</math>, one obtains the antiderivative of <math>\sin(x^\beta)/x^\alpha</math>.<ref>Victor Nijimbere, Ural Math J vol 3(1) and https://arxiv.org/abs/1703.01907 (2017)</ref>
The [[Lommel function]]<ref>Watson's "Treatise on the Theory of Bessel functions" (1966), Section 10.7, Equation (10)</ref> <math> s_{\mu, \nu} (z) = \frac{z^{\mu + 1}}{(\mu - \nu + 1)(\mu + \nu + 1)} {}_1F_2(1; \frac{\mu}{2} - \frac{\nu}{2} + \frac{3}{2} , \frac{\mu}{2} + \frac{\nu}{2} + \frac{3}{2} ;-\frac{z^2}{4}) </math>.
===The series <sub>2</sub>''F''<sub>0</sub>===
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