Special functions generated by rising and central factorial powers

T. P. Goy

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 2 (2016), pp. 19-32

Résumé

Replacing in the well-known series $\cos x=\sum\limits_{n=0}^\infty\frac{(-1)^nx^{2n}}{(2n)!}$, $\sin x=\sum\limits_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)!}$ falling factorial powers ($m!=m^{\underline{m}}$) by rising and central factorial powers ($m^{\overline{m}}$ and $m^{[m]}$ respectively), we obtain real functions $\mathrm{Cos}\, x=\sum\limits_{n=0}^\infty\frac{(-1)^nx^{2n}}{(2n)^{\overline{2n}}}$, $\mathrm{Sin}\, x=\sum\limits_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)^{\overline{2n+1}}}$, $\mathrm{Cosc}\, x=\sum\limits_{n=0}^\infty\frac{(-1)^nx^{2n}}{(2n)^{[2n]}}$, and $\mathrm{Sinc}\, x=\sum\limits_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)^{[2n+1]}}$. In this paper, we consider the non-elementary Fresnel-type integral functions $C_1(x)=\int\limits_0^x\mathrm{Cos}\,t^2 dt$, $S_1(x)=\int\limits_0^x\mathrm{Sin}\,t^2 dt$, $C_2(x)=\int\limits_0^x\mathrm{Cosc}\,t^2 dt$, $S_2(x)=\int\limits_0^x\mathrm{Sinc}\,t^2 dt$. We prove the following formulas: \begin{gather*} C_1(x)=4\left(\cos\frac{x^2}4 C\left(\frac x2\right)+\sin\frac{x^2}4 S\left(\frac x2\right)\right)-x,\\ S_1(x)=4\left(\sin\frac{x^2}4 C\left(\frac x2\right)-\cos\frac{x^2}4 S\left(\frac x2\right)\right),\\ C_2(x)=x-\frac{x^5}{20}{}_2F_3\left(1,\frac54;\frac43,\frac53,\frac94;-\frac{x^4}{27}\right),\quad S_2(x)=\frac{x^3}3{}_2F_3\left(\frac34,1;\frac56,\frac76,\frac74;-\frac{x^4}{27}\right), \end{gather*} where $C(p)$ and $S(p)$ are Fresnel integrals and $_2F_3(a_1,a_2;b_1,b_2,b_3;z)$ is a generalized hypergeometric function. We also show that functions $C_1(x)$, $S_1(x)$ are solutions of the ordinary linear second-order differential equations $4xy''-4y'+x^3y=-x^4-4$ and $4xy''-4y'+x^3y=4x^2$, respectively, and the functions $C_2(x)$, $S_2(x)$ are solutions of the ordinary linear fourth-order differential equations $27x^3y^{IV}-135x^2y'''+(16x^5+339)y''-384y'=-384$ and $27x^3y^{IV}-81x^2y'''+(16x^5+177x)y''+(32x^4-192)y'=0$, respectively.

Détail

Keywords: rising factorial power, central factorial power, Fresnel integrals, generalized hypergeometric function, Cauchy problem.