Fast Calculation of the Derivatives of Bessel Functions with Respect to the Parameter and Applications

Abstract

In this paper, the fast algorithms of the derivatives of Bessel functions with respect to the parameter are obtained. Based on these fast algorithms, we discuss the calculations of the derivatives of the functions related to the heterogeneous Bessel differential equation, such as Anger, Weber, Struve and modified Struve functions. In addition, the fast calculation of some integrals related to these functions are obtained. At last, numerical examples show the algorithms given in this paper are fast and high precision.

1. Introduction

Bessel equation is one of the common differential equations in mathematical physics. It has Noether symmetries under certain conditions. Moreover, the Bessel functions play a very important role in wave problems and various problems involving potential field. The most typical problems include electromagnetic wave propagation in cylindrical waveguide, heat conduction in cylinder and vibration modal analysis of circular (or annular) thin films. Moreover, the Bessel functions have had many applications such as the definition of FM synthesis in signal processing or Kaiser window.

The Bessel functions were first defined by Daniel Bernoulli. The Bessel functions $y(x)$ are canonical solutions of the following differential equation in [1]

$$x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+(x^2-{\alpha }^2)y=0$$

(1)

For (1), there are the following two linearly independent solutions: $J_{\alpha }(x)$, $Y_{\alpha }(x)$, where $J_{\alpha }(x)$ is the Bessel function of the first kind and $Y_{\alpha }(x)$ is the Bessel function of the second kind.

For the modified Bessel’s equation in [1]

$$x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}-(x^2+{\alpha }^2)y=0$$

(2)

there are two linearly independent solutions: $I_{\alpha }(x)$, $K_{\alpha }(x)$, where $I_{\alpha }(x)$ is the modified Bessel function of the first kind and $K_{\alpha }(x)$ is the modified Bessel function of the second kind.

Bessel functions of the third kind (Hankel functions) are $H_{{}_{\alpha }}^{(1)}(x)=J_{\alpha }(x)+i{Y}_{\alpha }(x)$, $H_{{}_{\alpha }}^{(2)}(x)=J_{\alpha }(x)-i{Y}_{\alpha }(x)$. For $J_{\alpha }(x)$, $I_{\alpha }(x)$, $Y_{\alpha }(x)$, $K_{\alpha }(x)$ in [2], there are

$$\left\{\begin{array}{c}{J}_{\alpha }(x)\\ I_{\alpha }(x)\end{array}\right\}={(\frac{x}{2})}^{\alpha }{}_0\widehat{F}{}_1(\alpha +1,1,\frac{\mp x^2}{4})$$

(3)

where

$${}_0\widehat{F}{}_1(\alpha ,\beta ,x)={\displaystyle \sum _{n=0}^{\infty }\frac{x^n}{\mathrm{\Gamma }(n+\beta )\mathrm{\Gamma }(n+\alpha )}}$$

(4)

and

$$\left\{\begin{array}{c}{Y}_{\alpha }(x)\\ K_{\alpha }(x)\end{array}\right\}=\frac{1}{\sin (\alpha \pi )}\left\{\begin{array}{c}{J}_{\alpha }(x)\cos (\alpha \pi )-J_{-\alpha }(x)\\ \frac{\pi }{2}(I_{-\alpha }(x)-I_{\alpha }(x))\end{array}\right\}$$

(5)

Recently, the study on Bessel functions and Bessel polynomials has attracted much attention. Riyasat et al. introduced a generalized 2D extension of q-Bessel polynomials and obtained orthogonality associated with Bessel-type Sheffffer sequences with q-parameters in [3,4]. Furthermore, some scholars have discussed analytical properties of the derivative of Bessel functions with respect to the parameters $\alpha$ in [5,6,7,8,9,10,11,12,13,14,15]. For example, the results of [11] were as follows:

$${\frac{\partial }{\partial \alpha }[J_{\alpha }(x)]|}_{\alpha =\frac{1}{2}}=J_{\frac{1}{2}}(x)\mathrm{C}\mathrm{i}(2x)-J_{-\frac{1}{2}}(x)\mathrm{S}\mathrm{i}(2x)$$

(6)

$${\frac{\partial }{\partial \alpha }[J_{\alpha }(x)]|}_{\alpha =-\frac{1}{2}}=J_{-\frac{1}{2}}(x)\mathrm{C}\mathrm{i}(2x)+J_{\frac{1}{2}}(x)\mathrm{S}\mathrm{i}(2x)$$

(7)

where Si(2x) and Ci(2x) are the sine and cosine integrals, respectively.

Moreover, Brychkov gave the following results in [5]:

$$\begin{gathered} \frac{\partial }{\partial \alpha }[J_{\alpha }(x)]=\frac{\pi (\frac{x}{2}{)}^{2\alpha }(Y_{\alpha }(x)-\cot (\alpha \pi )J_{\alpha }(x))}{2{\mathrm{\Gamma }}^2(\alpha +1)}{}_{2}F{}_3\left(\begin{array}{c}\alpha ,\alpha +\frac{1}{2};-x^2\\ \alpha +1,\alpha +1,2\alpha +1\end{array}\right) \\ +J_{\alpha }(x)\left[\frac{1}{2\alpha }-\psi (\alpha +1)+\ln \frac{x}{2}+\frac{x^2{J}_{\alpha }(x)}{4({\alpha }^2-1)}{}_{3}F{}_4\left(\begin{array}{c}1,1,\frac{3}{2};-x^2\\ 2,2,2-\alpha ,\alpha +2\end{array}\right)\right] \end{gathered}$$

(8)

and

$$\begin{gathered} \frac{{\partial }^m}{\partial {\alpha }^m}\left[\left\{\begin{array}{c}{J}_{\alpha }(x)\\ I_{\alpha }(x)\end{array}\right\}\right]={(-1)}^m\frac{m!i}{2\pi }{\displaystyle \sum _{k=0}^m\frac{1}{(m-k)!}}{\displaystyle \sum _{p=0}^{m-k}{(-1)}^p}\left(\begin{array}{c}m-k\\ p\end{array}\right) \\ \times \left[e^{i\alpha \pi }{(\ln \frac{z}{2}+\pi i)}^p-e^{-i\alpha \pi }{(\ln \frac{z}{2}-\pi i)}^p\right]{\mathrm{\Gamma }}^{(m-k-p)}(-\alpha ) \\ \times \{{\delta }_{k,0}\mathrm{\Gamma }(\alpha +1)\left\{\begin{array}{c}{J}_{\alpha }(x)\\ I_{\alpha }(x)\end{array}\right\}\mp \frac{(1-{\delta }_{k,0}){(z/2)}^{\alpha +2}}{{(\alpha +1)}^{k+1}} \\ \times F_{1:k+1;0}^{0:k+1;0}\left(\begin{array}{c}-:\alpha ,\alpha ,\cdots ,\alpha ;-;\\ 2:\alpha +1,\alpha +1,\cdots ,\alpha +1;-;\end{array}\pm \frac{x^2}{4},\mp \frac{x^2}{4}\right)\} \end{gathered}$$

(9)

where

$$(a_p)=a_1,a_2,\cdots ,a_p,{[a_p]}_k={\displaystyle \prod _{i=1}^p(a_i}{)}_k$$

and

$$F_{s:t;u}^{p:q;r}\left(\begin{array}{c}(a_p):(b_q);(c_r);\\ (d_s):(e_t);(f_u);\end{array}\omega ,z\right)={\displaystyle \sum _{j=0}^{\infty }{\displaystyle \sum _{k=0}^{\infty }\frac{{[a_p]}_{j+k}{[b_q]}_j{[c_r]}_k}{{[d_s]}_{j+k}{[e_t]}_j{[f_u]}_k}}}\frac{{\omega }^j{z}^k}{j!k!}$$

(10)

is the Kampe Feriet function.

Using the Leibniz’s law of derivation for (1), we have

$$x^2\frac{d^2{Z}_m}{d{x}^2}+x\frac{d{Z}_m}{dx}+(x^2-{\alpha }^2)Z_m=2m\alpha Z_{m-1}+m(m-1)Z_{m-2}$$

(11)

where $Z_m=\frac{{\partial }^{m}y}{\partial {\alpha }^m}$.

Therefore, $y=\frac{{\partial }^m}{\partial {\alpha }^m}[J_{\alpha }(x)]$ is a solution of the following inhomogeneous equation

$$x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+(x^2-{\alpha }^2)y=G^{(m)}(x)$$

(12)

where

$$G^{(m)}(x)=2m\alpha \frac{{\partial }^{m-1}}{\partial {\alpha }^{m-1}}[J_{\alpha }(x)]+m(m-1)\frac{{\partial }^{m-2}}{\partial {\alpha }^{m-2}}[J_{\alpha }(x)]$$

and the general solution of (11) is

$$C_1{J}_{\alpha }(x)+C_2{Y}_{\alpha }(x)+\frac{{\partial }^m}{\partial {\alpha }^m}[J_{\alpha }(x)]$$

(13)

Hence, the derivatives with respect to parameter of the Bessel function are closely related to inhomogeneous Bessel’s differential equation. It is clear that (9) cannot be used for the fast calculation of $\frac{{\partial }^m}{\partial {\alpha }^m}[J_{\alpha }(x)]$ and $\frac{{\partial }^m}{\partial {\alpha }^m}[I_{\alpha }(x)]$ for $m=1,2,\cdots$. In this article, we will consider the fast calculation of the derivatives of Bessel functions with respect to the parameter.

The structure of this paper is organized as follows. In Section 2, we obtain the recurrence formula of the derivative with respect to the parameter $\alpha$ of $J_{\alpha }(x)$ and $I_{\alpha }(x)$ and their applications. In Section 3, we obtain fast calculation of the derivative of $Y_{\alpha }(x),K_{\alpha }(x),H_{\alpha }^{(1)}(x),H_{\alpha }^{(2)}(x)$ and the correlation function with respect to the parameter and their applications. In Section 4, the conclusion is given.

2. The Recurrence Formula of the Derivative with Respect to the Parameter $\alpha$ of $J_{\alpha }(x)$ and $I_{\alpha }(x)$ and Their Applications

In this section, we always assume that $\alpha \ne -1,-2,\cdots$ and $m=1,2,\cdots$.

If we can obtain the fast algorithm of $\frac{{\partial }^m}{\partial {\alpha }^m}[J_{\alpha }(x)]$, then we will obtain the fast algorithm of $\frac{{\partial }^m}{\partial {\alpha }^m}[I_{\alpha }(x)]$ by $I_{\alpha }(x)=e^{-\frac{\alpha \pi i}{2}}{J}_{\alpha }(e^{\frac{\pi i}{2}}x)$.

In the following, we will consider the fast algorithm for $\frac{{\partial }^m}{\partial {\alpha }^m}[J_{\alpha }(x)]$ and $\frac{{\partial }^m}{\partial {\alpha }^m}[I_{\alpha }(x)]$.

First, the following results were given in [2]:

$$\frac{\partial }{\partial \alpha }[J_{\alpha }(x)]=J_{\alpha }(x)\ln \frac{x}{2}-(\frac{x}{2}{)}^{\alpha }\sum _{n=0}^{\infty }\frac{{(-1)}^n\psi (n+\alpha +1)}{\mathrm{\Gamma }(n+\alpha +1)n!}{(\frac{x^2}{4})}^n$$

(14)

and

$$\frac{\partial }{\partial \alpha }[Y_{\alpha }(x)]=\cot (\alpha \pi )(\frac{\partial J_{\alpha }(x)}{\partial \alpha }-\pi Y_{\alpha }(x))-\csc (\alpha \pi )\frac{\partial }{\partial \alpha }[J_{-\alpha }(x)]-\pi J_{\alpha }(x)$$

(15)

where $\psi (\alpha )$ is digamma function defined by

$$\psi (\alpha )=-\gamma +{\displaystyle \sum _{n=0}^{\infty }(\frac{1}{n+1}}-\frac{1}{n+\alpha })$$

(16)

and $\gamma$ is the Euler-Mascheroni constant.

Using the Leibniz’s law of derivation for (3), we have the following result:

$$\frac{{\partial }^m}{\partial {\alpha }^m}\left\{\begin{array}{c}{J}_{\alpha }(x)\\ I_{\alpha }(x)\end{array}\right\}=(\frac{x}{2}{)}^{\alpha }{\displaystyle \sum _{i=0}^m\left(\begin{array}{c}m\\ i\end{array}\right)}{\ln }^{m-i}\frac{x}{2}{\displaystyle \sum _{n=0}^{\infty }\frac{{\mathrm{\Gamma }}_{-}^{(i)}(n+\alpha +1)}{n!}(\frac{\mp x^2}{4}}{)}^n$$

(17)

where

$${\mathrm{\Gamma }}_{+}(\alpha )=\mathrm{\Gamma }(\alpha ),{\mathrm{\Gamma }}_{-}(\alpha )=\frac{1}{\mathrm{\Gamma }(\alpha )}$$

(18)

For ${\mathrm{\Gamma }}_{\pm }^{(i)}(\alpha )$, the following recurrence has been obtained in [16].

Lemma 1.

If$\alpha \ne 0,-1,-2,\cdots$and$m=1,2,\cdots$, then

$${\mathrm{\Gamma }}_{\pm }^{(m)}(\alpha )={\mathrm{\Gamma }}_{\pm }(\alpha )H_m^{{\psi }_{\pm }}(\alpha )$$

(19)

where

$$\begin{gathered} H_0^{{\psi }_{\pm }}(\alpha )=1 \\ {H}_l^{{\psi }_{\pm }}(\alpha )={\displaystyle \sum _{k=0}^{l-1}\left(\begin{array}{c}l-1\\ k\end{array}\right)}{\psi }_{\pm }^{(k)}(\alpha )H_{l-1-k}^{{\psi }_{\pm }}(\alpha ),l=1,2,\cdots ,m \\ {\psi }_{\pm }(\alpha )=\pm \psi (\alpha ) \end{gathered}$$

(20)

Moreover, by

$$\begin{gathered} {\psi }^{(k)}(x)=k!{(-1)}^{k+1}\zeta (k+1,x), \\ \zeta (s,n+x)=\zeta (s,x)-{\displaystyle \sum _{l=0}^{n-1}\frac{1}{{(l+x)}^s}} \\ \zeta (s,-n+x)=\zeta (s,x)+{\displaystyle \sum _{l=1}^n\frac{1}{{(x-l)}^s}} \\ \zeta (s,\frac{1}{2})=(2^s-1)\zeta (s) \end{gathered}$$

(21)

and

$$\zeta (k,1)=\zeta (k,0)=\{\begin{array}{c}\gamma ,k=1\\ \zeta (k),k>1\end{array}$$

(22)

$$\zeta (k,\frac{1}{2})=\{\begin{array}{c}\gamma +2\ln 2,k=1\\ (2^k-1)\zeta (k),k>1\end{array}$$

(23)

see [17], we have the following lemma.

Lemma 2.

If$x\ne -1,-2,\cdots$and$n=0,1,2,\cdots$, then

$${[{\mathrm{\Gamma }}_{\pm }(x+n)]}^{(k)}={\mathrm{\Gamma }}_{\pm }(x+n)H_{\pm ,n,k}(x)$$

(24)

where

$$H_{\pm ,n,0}(x)=1,H_{\delta ,n,j}(x)=\mp {\displaystyle \sum _{i=0}^{j-1}{(1-j)}_i}{R}_{i,n}(x)H_{\pm ,n,j-1-i}(x)$$

(25)

$$R_{i,0}(x)=\zeta (i+1,x),R_{i,n}(x)=R_{i,n-1}(x)-\frac{1}{{(n+x-1)}^{i+1}}$$

(26)

for$i=0,1,2,\cdots ,k.$

By Lemma 2, we obtain the following lemma.

Lemma 3.

If$\alpha ,\beta \ne -1,-2,\cdots$and$m,n=1,2,\cdots$, then

$$\frac{{\partial }^{m+n}}{\partial {\alpha }^m\partial {\beta }^n}[{}_0\widehat{F}{}_1(\alpha ,\beta ,x)]={}_0\widehat{F}{}_1^{(m,n)}(\alpha ,\beta ,x)={\displaystyle \sum _{k=0}^{\infty }\frac{H_{-1,k,m}(\alpha )H_{-1,k,n}(\beta )x^n}{\mathrm{\Gamma }(k+\alpha )\mathrm{\Gamma }(k+\beta )}}$$

(27)

In the following, we obtain the fast algorithm of $\frac{{\partial }^m}{\partial {\alpha }^m}[J_{\alpha }(x)]$ and $\frac{{\partial }^m}{\partial {\alpha }^m}[I_{\alpha }(x)]$ by Lemma 3 and (17).

Theorem 1.

If$x\ne -1,-2,\cdots$and$m=1,2,\cdots$, then

$$\frac{{\partial }^m}{\partial {\alpha }^m}\left\{\begin{array}{c}{J}_{\alpha }(x)\\ I_{\alpha }(x)\end{array}\right\}=(\frac{x}{2}{)}^{\alpha }{\displaystyle \sum _{i=0}^m\left(\begin{array}{c}m\\ i\end{array}\right)}{\ln }^{m-i}\frac{x}{2}{}_0\widehat{F}{}_1^{(i,0)}(\alpha +1,1,\frac{\mp x^2}{4})$$

(28)

To verify the effectiveness of (28), we will verify it in Mathematica.

In the following, we can use Symbolic computation and Numerical integration for calculating $\frac{{\partial }^m}{\partial {\alpha }^m}\left\{\begin{array}{c}{J}_{\alpha }(x)\\ I_{\alpha }(x)\end{array}\right\}$ in Mathematica, respectively.

(1)

Symbolic computation.

N[D[BesslJ[alphah,x],{alphah,m}]/.alphah−> alpha,prec]

(29)

and

N[D[BesslI[alphah,x],{alphah,m}]/.alphah−> alpha,prec]

(30)

where prec is the specified number of calculated precision bits.

(2)

Numerical integration.

By the following integral representations of $J_{\alpha }(x)$ and $I_{\alpha }(x)$ in [2],

$$J_{\alpha }(x)=\frac{1}{\pi }{\displaystyle {\int }_0^{\pi }\cos (\alpha \theta -x\sin \theta )d\theta -\frac{\sin (\alpha \pi )}{\pi }}{\displaystyle {\int }_0^{\infty }{e}^{-x\sinh t-\alpha t}}dt$$

(31)

and

$$I_{\alpha }(x)=\frac{1}{\pi }{\displaystyle {\int }_0^{\pi }{e}^{x\cos \theta }\cos (\alpha \theta )d\theta -\frac{\sin (\alpha \pi )}{\pi }}{\displaystyle {\int }_0^{\infty }{e}^{-x\cosh t-\alpha t}}dt$$

(32)

we have

$$\begin{gathered} \frac{{\partial }^m}{\partial {\alpha }^m}[J_{\alpha }(x)]=\frac{1}{\pi }{\displaystyle {\int }_0^{\pi }{\theta }^m}\cos (\alpha \theta -x\sin \theta +\frac{m\pi }{2})d\theta \\ -{\displaystyle {\int }_0^{\infty }{\displaystyle \sum _{l=0}^{\infty }\left(\begin{array}{c}m\\ l\end{array}\right)}{(-1)}^l}{\pi }^{m-l-1}\sin (\alpha \pi +\frac{(m-l)\pi }{2})t^l{e}^{-x\sinh t-\alpha t}dt \end{gathered}$$

(33)

and

$$\begin{gathered} \frac{{\partial }^m}{\partial {\alpha }^m}[I_{\alpha }(x)]=\frac{1}{\pi }{\displaystyle {\int }_0^{\pi }{\theta }^m{e}^{x\cos \theta }}\cos (\alpha \theta +\frac{m\pi }{2})d\theta \\ -{\displaystyle {\int }_0^{\infty }{\displaystyle \sum _{l=0}^{\infty }\left(\begin{array}{c}m\\ l\end{array}\right)}{(-1)}^l}{\pi }^{m-l-1}\sin (\alpha \pi +\frac{(m-l)\pi }{2})t^l{e}^{-x\cosh t-\alpha t}dt. \end{gathered}$$

(34)

In Mathematica, (33) and (34) can be calculated by internal numerical integral function. Some numerical results are as follows.

Where $(28)J$ and $(28)I$ denote the calculation of $\frac{{\partial }^m}{\partial {\alpha }^m}[J_{\alpha }(x)]$ and $\frac{{\partial }^m}{\partial {\alpha }^m}[I_{\alpha }(x)]$ by (28), respectively. From Table 1, we see that (29) and (30) have the same computational precision as (28). However, the calculating time of (29) and (30) is time-consuming. Moreover, (33) and (34) not only have the big error but also quite time-consuming. Therefore, (28) is an effective algorithm due to its high accuracy and fast computation.

Table 1. The comparison of the numerical results for $\frac{{\partial }^m}{\partial {\alpha }^m}[J_{\alpha }(x)]$ and $\frac{{\partial }^m}{\partial {\alpha }^m}[I_{\alpha }(x)]$.

Algorithm	x, α, m	Time16, Err	Time32, Err	Time48, Err	Value
(28) J	$$\frac{17}{4},\frac{16}{3},5$$	0.000000,10−17	0.000000,10−37	0.015625,10−57	0.099929488104248589368253077762 094775507915325708…
(28) I	$$\frac{17}{4},\frac{16}{3},5$$	0.000000,10−17	0.000000,10−37	0.000000,10−57	0.072368397550548512467138097425 131708648435197397…
(29)	$$\frac{17}{4},\frac{16}{3},5$$	7.875000,10−16	25.593750,10−32	123.546875,10−48
(30)	$$\frac{17}{4},\frac{16}{3},5$$	8.312500,10−16	27.531250,10−32	125.046875,10−48
(33)	$$\frac{17}{4},\frac{16}{3},5$$	0.031250,10−8	0.046875,10−29	0.093750,10−47
(34)	$$\frac{17}{4},\frac{16}{3},5$$	0.062500,10−13	0.109375,10−29	0.156250,10−48

Furthermore, by Lemma 1 and the Leibniz’s law of derivation we have the following lemma.

Lemma 4.

If$\alpha \ne -1,-2,\cdots$and$m=1,2,\cdots$, then

$${(x^{\alpha }{\mathrm{\Gamma }}_{\pm }(a\alpha +b))}^{(m)}=x^{\alpha }{\mathrm{\Gamma }}_{\pm }(a\alpha +b)P_{m,\alpha }^{+}(x,a,b)$$

(35)

where

$$P_{m,\alpha }^{+}(x,a,b)={\displaystyle \sum _{l=0}^m\left(\begin{array}{c}m\\ l\end{array}\right)}{a}^l{H}_l^{{\psi }_{\pm }}(a\alpha +b){\ln }^{m-l}x$$

(36)

In the following, we give some integrals related to $J_{\alpha }(x)$ and $I_{\alpha }(x)$. The results of [1] and [2] are as follows:

$$\begin{gathered} J_{\alpha }(x)=\frac{{(\frac{x}{2})}^{\alpha }}{\sqrt{\pi }\mathrm{\Gamma }(\alpha +\frac{1}{2})}{\displaystyle {\int }_0^{\pi }{(\sin \theta )}^{2\alpha }\cos (x\cos \theta )d\theta }[Re\alpha >-\frac{1}{2}] \\ =\frac{2{(\frac{x}{2})}^{\alpha }}{\sqrt{\pi }\mathrm{\Gamma }(\alpha +\frac{1}{2})}{\displaystyle {\int }_0^1{(1-t^2)}^{\alpha -\frac{1}{2}}\cos (xt)dt}[Re\alpha >-\frac{1}{2}] \\ =\frac{2}{\sqrt{\pi }{(\frac{x}{2})}^{\alpha }\mathrm{\Gamma }(\frac{1}{2}-\alpha )}{\displaystyle {\int }_1^{\infty }\frac{\sin (xt)}{{(t^2-1)}^{\alpha +\frac{1}{2}}}dt}[\left|Re\alpha \right|<\frac{1}{2},x>0] \end{gathered}$$

(37)

$$\begin{gathered} I_{\alpha }(x)=\frac{{(\frac{x}{2})}^{\alpha }}{\sqrt{\pi }\mathrm{\Gamma }(\alpha +\frac{1}{2})}{\displaystyle {\int }_0^{\pi }{e}^{\pm x\cos \theta }{(\sin \theta )}^{2\alpha }d\theta }[Re\alpha >-\frac{1}{2}] \\ =\frac{{(\frac{x}{2})}^{\alpha }}{\sqrt{\pi }\mathrm{\Gamma }(\alpha +\frac{1}{2})}{\displaystyle {\int }_{-1}^1{(1-t^2)}^{\alpha -\frac{1}{2}}{e}^{\pm xt}dt}[Re\alpha >-\frac{1}{2}] \end{gathered}$$

(38)

Thus, we have

$$\begin{gathered} {\displaystyle {\int }_0^{\pi }{(\sin \theta )}^{2\alpha }\cos (x\cos \theta )d\theta }=2{\displaystyle {\int }_0^1{(1-t^2)}^{\alpha -\frac{1}{2}}\cos (xt)dt} \\ =\sqrt{\pi }(\frac{2}{x}{)}^{\alpha }\mathrm{\Gamma }(\alpha +\frac{1}{2})J_{\alpha }(x) \end{gathered}$$

(39)

and

$$\begin{gathered} {\displaystyle {\int }_0^{\pi }{2}^m{(\sin \theta )}^{2\alpha }\cos (x\cos \theta ){\ln }^m\sin \theta d\theta } \\ =2{\displaystyle {\int }_0^1{(1-t^2)}^{\alpha -\frac{1}{2}}\cos (xt){\ln }^m(1-t^2)dt} \\ =\sqrt{\pi }(\frac{2}{x}{)}^{\alpha }\mathrm{\Gamma }(\alpha +\frac{1}{2}){\displaystyle \sum _{l=0}^m\left(\begin{array}{c}m\\ l\end{array}\right)}{P}_{m-l,\alpha }^{+}(\frac{2}{x},1,\frac{1}{2})\frac{d^l}{d{\alpha }^l}[J_{\alpha }(x)] \end{gathered}$$

(40)

where $Re\alpha >-\frac{1}{2}.$

In the same way, we have

$$\begin{gathered} {\displaystyle {\int }_1^{\infty }\frac{\sin (xt)}{{(t^2-1)}^{\alpha +\frac{1}{2}}}{\ln }^m\frac{1}{t^2-1}}dt \\ =\frac{\sqrt{\pi }(\frac{x}{2}{)}^{\alpha }\mathrm{\Gamma }(\frac{1}{2}-\alpha )}{2}{\displaystyle \sum _{l=0}^m\left(\begin{array}{c}m\\ l\end{array}\right)}{P}_{m-l,\alpha }^{+}(\frac{x}{2},-1,\frac{1}{2})\frac{d^l}{d{\alpha }^l}[J_{\alpha }(x)] \end{gathered}$$

(41)

for $0<\alpha +\frac{1}{2}<1$ and

$$\begin{gathered} {\displaystyle {\int }_{-1}^1{(1-t^2)}^{\alpha -\frac{1}{2}}{e}^{xt}{\ln }^m(1-t^2)}dt \\ =\sqrt{\pi }(\frac{2}{x}{)}^{\alpha }\mathrm{\Gamma }(\alpha +\frac{1}{2}){\displaystyle \sum _{l=0}^m\left(\begin{array}{c}m\\ l\end{array}\right)}{P}_{m-l,\alpha }^{+}(\frac{2}{x},1,\frac{1}{2})\frac{d^l}{d{\alpha }^l}[I_{\alpha }(x)] \end{gathered}$$

(42)

for $Re\alpha >-\frac{1}{2}.$

For (40)–(42), we will compare the left section ${(\ast )}_L$ with their right expression ${(\ast )}_R$ of the formula $(\ast )$. The results are as follows.

It can be seen from Table 2 that the algorithm of ${(40)}_R$, ${(41)}_R$ and ${(42)}_R$ are fast and high precision. Numerical integration is not only time-consuming, but also cannot achieve the specified precision. Sometimes we can not obtain the correct results. For example, let $\alpha =\frac{1}{4}$, $x=\frac{3}{4}$, $m=5$ in (41), the results of numerical integration in Mathematica are 4966.858⋯, 7490.425⋯ and −16,078.679⋯ with the specified precision of 16, 32 and 48 bits, respectively. However, the accurate value is 167,540.81926⋯. Therefore, it is necessary for many singular integrals to find the corresponding expression of the special functions and their derivatives.

Table 2. The comparison of the numerical results for (40)–(42).

Algorithm	α,x,m	Time16,Err	Time32, Err	Time48, Err	Value
(40)L	$-\frac{1}{3},\frac{3}{4},4$	0.062500,10−7	0.203125,10−11	0.234375,10−12	136557.84577593821918 99633311486…
(40)R	$-\frac{1}{3},\frac{3}{4},4$	0.000000,10−20	0.000000,10−40	0.000000,10−60
(41)L	$-\frac{1}{4},\frac{3}{4},5$	0.062500,10−12	0.125000,10−19	0.187500,10−26	278.09550436403709 0207658787058492…
(41)R	$-\frac{1}{4},\frac{3}{4},5$	0.015625,10−19	0.000000,10−39	0.000000,10−59
(42)L	$-\frac{1}{5},\frac{25}{3},3$	0.031250,10−8	0.093750,10−14	0.140625,10−19	−1.526973433859057 03392895856…×106
(42)R	$-\frac{1}{5},\frac{25}{3},3$	0.000000,10−20	0.000000,10−40	0.000000,10−60

3. Fast Calculation of the Derivative of $Y_{\alpha }(x)$, $K_{\alpha }(x)$, $H_{\alpha }^{(1)}(x)$, $H_{\alpha }^{(2)}(x)$ and the Correlation Function with Respect to the Parameter and Their Applications

In this section, we will obtain the fast calculation of the derivative of $Y_{\alpha }(x)$, $K_{\alpha }(x)$, $H_{\alpha }^{(1)}(x)$, $H_{\alpha }^{(2)}(x)$ and the correlation function with respect to the parameter and their applications. Moreover, we always assume that $\alpha \ne 0,\pm 1,\pm 2,\cdots$ and $m=1,2,\cdots$ in this section.

By (5), we have

$$\begin{gathered} \sin (\alpha \pi )Y_{\alpha }(x)=(\cos (\alpha \pi )J_{\alpha }(x)-J_{-\alpha }(x)), \\ \sin (\alpha \pi )Y_{\alpha }(x)=\frac{\pi }{2}(I_{-\alpha }(x)-I_{\alpha }(x)), \end{gathered}$$

(43)

so we have the following recurrence formulas

$$\begin{array}{l}\frac{{\partial }^m}{\partial {\alpha }^m}[Y_{\alpha }(x)]\\ =\frac{1}{\sin (\alpha \pi )}({\displaystyle \sum _{l=0}^m\left(\begin{array}{c}m\\ l\end{array}\right)}{\pi }^{m-l}\cos \frac{(2\alpha +m-l)\pi }{2}\frac{{\partial }^l}{\partial {\alpha }^l}{J}_{\alpha }(x)-(-1{)}^m\frac{{\partial }^m}{\partial t^m}{J_t(x)|}_{t=-\alpha })\\ -\frac{1}{\sin (\alpha \pi )}{\displaystyle \sum _{l=0}^{m-1}\left(\begin{array}{c}m\\ l\end{array}\right)}{\pi }^{m-l}\sin \frac{(2\alpha +m-l)\pi }{2}\frac{{\partial }^l}{\partial {\alpha }^l}{Y}_{\alpha }(x)\end{array}$$

(44)

and

$$\begin{gathered} \begin{array}{l}\frac{{\partial }^m}{\partial {\alpha }^m}[K_{\alpha }(x)]\\ =\frac{\pi }{2\sin (\alpha \pi )}((-1{)}^m\frac{{\partial }^m}{\partial t^m}{I_t(x)|}_{t=-\alpha }-\frac{{\partial }^m}{\partial {\alpha }^m}{I}_{\alpha }(x))\end{array} \\ -\frac{1}{\sin (\alpha \pi )}{\displaystyle \sum _{l=0}^{m-1}\left(\begin{array}{c}m\\ l\end{array}\right)}{\pi }^{m-l}\sin \frac{(2\alpha +m-l)\pi }{2}\frac{{\partial }^l}{\partial {\alpha }^l}{K}_{\alpha }(x). \end{gathered}$$

(45)

For the derivative of Hankel function $H_{\alpha }^{(1)}(x)$, $H_{\alpha }^{(2)}(x)$ with respect to the parameter, we have

$$\begin{gathered} \frac{{\partial }^m}{\partial {\alpha }^m}[H_{\alpha }^{(1)}(x)]=\frac{{\partial }^m}{\partial {\alpha }^m}[J_{\alpha }(x)]+i\frac{{\partial }^m}{\partial {\alpha }^m}[Y_{\alpha }(x)] \\ \frac{{\partial }^m}{\partial {\alpha }^m}[H_{\alpha }^{(2)}(x)]=\frac{{\partial }^m}{\partial {\alpha }^m}[J_{\alpha }(x)]-i\frac{{\partial }^m}{\partial {\alpha }^m}[Y_{\alpha }(x)] \end{gathered}$$

(46)

where $\frac{{\partial }^m}{\partial {\alpha }^m}[J_{\alpha }(x)]$ and $\frac{{\partial }^m}{\partial {\alpha }^m}[Y_{\alpha }(x)]$ are calculated according to (28) and (44), respectively.

In the following, we will consider the derivatives of the functions related to the inhomogeneous Bessel equation with respect to parameter. The Anger function ${\mathrm{J}}_{\mathsf{\alpha }}(\mathrm{x})$ satisfies the following equation in [2]

$$x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+(x^2-{\alpha }^2)y=\frac{(z-\alpha )\sin (\alpha \pi )}{\pi }$$

(47)

and it has the following integral representation and the series expansion

$$\begin{gathered} {\mathrm{J}}_{\mathsf{\alpha }}(\mathrm{x})=\frac{1}{\pi }{\displaystyle {\int }_0^{\pi }\cos (\alpha \theta -x\sin \theta )d\theta } \\ =\cos \frac{\alpha \pi }{2}{}_0\widehat{F}{}_1(1-\frac{\alpha }{2},1+\frac{\alpha }{2},\frac{-x^2}{4})+\frac{x}{2}\sin \frac{\alpha \pi }{2}{}_0\widehat{F}{}_1(\frac{3-\alpha }{2},\frac{3+\alpha }{2},\frac{-x^2}{4}) \end{gathered}$$

(48)

Moreover, the Weber function ${\mathrm{E}}_{\mathsf{\alpha }}(\mathrm{x})$ satisfies the following equation in [2]

$$x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+(x^2-{\alpha }^2)y=-\frac{x+\alpha +(z-\alpha )\cos (\alpha \pi )}{\pi }$$

(49)

and it has the following integral representation and the series expansion

$$\begin{gathered} {\mathrm{E}}_{\mathsf{\alpha }}(\mathrm{x})=\frac{1}{\pi }{\displaystyle {\int }_0^{\pi }\sin (\alpha \theta -x\sin \theta )d\theta } \\ =\sin \frac{\alpha \pi }{2}{}_0\widehat{F}{}_1(1-\frac{\alpha }{2},1+\frac{\alpha }{2},\frac{-x^2}{4})-\frac{x}{2}\cos \frac{\alpha \pi }{2}{}_0\widehat{F}{}_1(\frac{3-\alpha }{2},\frac{3+\alpha }{2},\frac{-x^2}{4}) \end{gathered}$$

(50)

The Struve function ${\mathrm{H}}_{\mathsf{\alpha }}(\mathrm{x})$ has the following power series form in [2]

$${\mathrm{H}}_{\mathsf{\alpha }}(\mathrm{x})={(\frac{x}{2})}^{\alpha +1}{}_0\widehat{F}{}_1(\frac{3}{2}+\alpha ,\frac{3}{2},\frac{-x^2}{4})$$

(51)

and it is a solution of the non-homogeneous Bessel’s differential equation

$$x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+(x^2-{\alpha }^2)y=\frac{4}{\sqrt{\pi }\mathrm{\Gamma }(\alpha +\frac{1}{2})}{(\frac{x}{2})}^{\alpha +1}$$

(52)

The modified Struve function ${\mathrm{L}}_{\mathsf{\alpha }}(\mathrm{x})$ has the following power series form in [2]

$${\mathrm{L}}_{\alpha }(\mathrm{x})={(\frac{x}{2})}^{\alpha +1}{}_0\widehat{F}{}_1(\frac{3}{2}+\alpha ,\frac{3}{2},\frac{x^2}{4})$$

(53)

and it is a solution of the non-homogeneous Bessel’s differential equation

$$x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}-(x^2+{\alpha }^2)y=\frac{4}{\sqrt{\pi }\mathrm{\Gamma }(\alpha +\frac{1}{2})}{(\frac{x}{2})}^{\alpha +1}.$$

(54)

By Lemma 3 and (48), we have the following theorem.

Theorem 2.

If$x\ne -1,-2,\cdots$and$m=1,2,\cdots$, then

$$\begin{gathered} \frac{{\partial }^m}{\partial {\alpha }^m}{\mathrm{J}}_{\mathsf{\alpha }}(x)={\displaystyle \sum _{l=0}^m\left(\begin{array}{c}m\\ l\end{array}\right)}\frac{{\pi }^{m-l}}{2^m}\cos \frac{(\alpha +m-l)\pi }{2}{\displaystyle \sum _{i=0}^l\left(\begin{array}{c}l\\ i\end{array}\right)}{(-1)}^i{}_0\widehat{F}{}_1^{(i,l-i)}(\frac{2-\alpha }{2},\frac{2+\alpha }{2},\frac{-x^2}{4}) \\ +\frac{x}{2}{\displaystyle \sum _{l=0}^m\left(\begin{array}{c}m\\ l\end{array}\right)}\frac{{\pi }^{m-l}}{2^m}\sin \frac{(\alpha +m-l)\pi }{2}{\displaystyle \sum _{i=0}^l\left(\begin{array}{c}l\\ i\end{array}\right)}{(-1)}^i{}_0\widehat{F}{}_1^{(i,l-i)}(\frac{3-\alpha }{2},\frac{3+\alpha }{2},\frac{-x^2}{4}) \end{gathered}$$

(55)

$$\begin{gathered} \frac{{\partial }^m}{\partial {\alpha }^m}{\mathrm{E}}_{\mathsf{\alpha }}(x)={\displaystyle \sum _{l=0}^m\left(\begin{array}{c}m\\ l\end{array}\right)}\frac{{\pi }^{m-l}}{2^m}\sin \frac{(\alpha +m-l)\pi }{2}{\displaystyle \sum _{i=0}^l\left(\begin{array}{c}l\\ i\end{array}\right)}{(-1)}^i{}_0\widehat{F}{}_1^{(i,l-i)}(\frac{2-\alpha }{2},\frac{2+\alpha }{2},\frac{-x^2}{4}) \\ -\frac{x}{2}{\displaystyle \sum _{l=0}^m\left(\begin{array}{c}m\\ l\end{array}\right)}\frac{{\pi }^{m-l}}{2^m}\cos \frac{(\alpha +m-l)\pi }{2}{\displaystyle \sum _{i=0}^l\left(\begin{array}{c}l\\ i\end{array}\right)}{(-1)}^i{}_0\widehat{F}{}_1^{(i,l-i)}(\frac{3-\alpha }{2},\frac{3+\alpha }{2},\frac{-x^2}{4}) \end{gathered}$$

(56)

$$\frac{{\partial }^m}{\partial {\alpha }^m}{\mathrm{H}}_{\mathsf{\alpha }}(x)={(\frac{x}{2})}^{\alpha +1}{\displaystyle \sum _{l=0}^m\left(\begin{array}{c}m\\ i\end{array}\right)}{\ln }^{m-i}\frac{x}{2}{}_0\widehat{F}{}_1^{(i,0)}(\frac{3}{2}+\alpha ,\frac{3}{2},\frac{-x^2}{4})$$

(57)

$$\frac{{\partial }^m}{\partial {\alpha }^m}{\mathrm{L}}_{\mathsf{\alpha }}(x)={(\frac{x}{2})}^{\alpha +1}{\displaystyle \sum _{l=0}^m\left(\begin{array}{c}m\\ i\end{array}\right)}{\ln }^{m-i}\frac{x}{2}{}_0\widehat{F}{}_1^{(i,0)}(\frac{3}{2}+\alpha ,\frac{3}{2},\frac{x^2}{4})$$

(58)

Moreover, some integrals related to $Y_{\alpha }(x)$, $K_{\alpha }(x)$, ${\mathrm{J}}_{\mathsf{\alpha }}(x)$, ${\mathrm{E}}_{\mathsf{\alpha }}(x)$, ${\mathrm{H}}_{\mathsf{\alpha }}(x)$ and ${\mathrm{L}}_{\mathsf{\alpha }}(x)$ are given in [1] and [2]. For example,

$$\begin{gathered} Y_{\alpha }(z)=-\frac{2{(\frac{2}{z})}^{\alpha }}{\sqrt{\pi }\mathrm{\Gamma }(\frac{1}{2}-\alpha )}{\displaystyle {\int }_1^{\infty }\frac{\cos (zt)}{{(t^2-1)}^{\alpha +\frac{1}{2}}}dt}[z>0,-\frac{1}{2}<Re\alpha <\frac{1}{2}] \\ =-\frac{2^{\alpha +1}{z}^{\alpha }}{\sqrt{\pi }\mathrm{\Gamma }(\alpha +\frac{1}{2})}{\displaystyle {\int }_0^{\frac{\pi }{2}}\frac{{\cos }^{\alpha -\frac{1}{2}}t\cos (z-\alpha t+\frac{t}{2})}{{\sin }^{2\alpha +1}t}{e}^{-2z\cot t}dt}[\left|\arg z\right|<\frac{\pi }{2},Re(\alpha +\frac{1}{2})>0] \end{gathered}$$

(59)

$$K_{\alpha }(z)=\frac{\sqrt{\pi }{(\frac{z}{2})}^{\alpha }}{\mathrm{\Gamma }(\frac{1}{2}+\alpha )}{\displaystyle {\int }_1^{\infty }{e}^{-zt}{(t^2-1)}^{\alpha -\frac{1}{2}}dt}[\left|\arg z\right|<\frac{\pi }{2},Re\alpha >-\frac{1}{2}]$$

(60)

$$K_{\alpha }(xz)=\frac{\mathrm{\Gamma }(\frac{1}{2}+\alpha ){(2z)}^{\alpha }}{\sqrt{\pi }{x}^{\alpha }}{\displaystyle {\int }_0^{\infty }\frac{\cos (xt)}{{(t^2+z^2)}^{\alpha +\frac{1}{2}}}dt}[x>0,\left|\arg z\right|<\frac{\pi }{2},Re\alpha >-\frac{1}{2}]$$

(61)

$$\left\{\begin{array}{c}{\mathrm{H}}_{\mathsf{\alpha }}(x)\\ {\mathrm{L}}_{\mathsf{\alpha }}(x)\end{array}\right\}=\frac{2{(\frac{x}{2})}^{\alpha }}{\sqrt{\pi }\mathrm{\Gamma }(\frac{1}{2}+\alpha )}{\displaystyle {\int }_0^1{(1-t^2)}^{\alpha -\frac{1}{2}}}\left\{\begin{array}{c}\sin (xt)\\ \sinh (xt)\end{array}\right\}dt[Re\alpha >-\frac{1}{2}]$$

(62)

and

$$H_{\alpha }^{(1)}(z)=-\frac{2i{(\frac{2}{z})}^{\alpha }}{\sqrt{\pi }\mathrm{\Gamma }(\frac{1}{2}-\alpha )}{\displaystyle {\int }_1^{\infty }\frac{e^{izt}}{{(t^2-1)}^{\alpha +\frac{1}{2}}}}dt[-\frac{1}{2}<Re\alpha <\frac{1}{2},x>0]$$

(63)

Using (59)–(63), (48) and (50), many integrals can be expressed according to the derivatives of related Bessel functions with respect to the parameter. We obtain the following results.

By (59), we have

$$\begin{gathered} {\displaystyle {\int }_1^{\infty }\frac{\cos (xt)}{{(t^2-1)}^{\alpha +\frac{1}{2}}}{\ln }^m\frac{1}{t^2-1}dt} \\ =-\frac{{(\frac{x}{2})}^{\alpha }\sqrt{\pi }\mathrm{\Gamma }(\frac{1}{2}-\alpha )}{2}{\displaystyle \sum _{l=0}^m\left(\begin{array}{c}m\\ l\end{array}\right)}{P}_{m-l,\alpha }^{+}(\frac{x}{2},-1,\frac{1}{2})\frac{{\partial }^l}{\partial {\alpha }^l}[Y_{\alpha }(x)] \end{gathered}$$

(64)

By (60), we have

$$\begin{gathered} {\displaystyle {\int }_1^{\infty }{e}^{-xt}{(t^2-1)}^{\alpha -\frac{1}{2}}{\ln }^m(t^2-1)dt} \\ =\frac{{(\frac{2}{x})}^{\alpha }\mathrm{\Gamma }(\frac{1}{2}+\alpha )}{\sqrt{\pi }}{\displaystyle \sum _{l=0}^m\left(\begin{array}{c}m\\ l\end{array}\right)}{P}_{m-l,\alpha }^{+}(\frac{2}{x},1,\frac{1}{2})\frac{{\partial }^l}{\partial {\alpha }^l}[K_{\alpha }(x)] \end{gathered}$$

(65)

By (61), we have

$$\begin{gathered} {\displaystyle {\int }_0^{\infty }\frac{\cos (xt)}{{(t^2+z^2)}^{\alpha +\frac{1}{2}}}{\ln }^m\frac{1}{t^2+z^2}dt} \\ =\frac{\sqrt{\pi }{(\frac{x}{2z})}^{\alpha }}{\mathrm{\Gamma }(\frac{1}{2}+\alpha )}{\displaystyle \sum _{l=0}^m\left(\begin{array}{c}m\\ l\end{array}\right)}{P}_{m-l,\alpha }^{-}(\frac{x}{2z},1,\frac{1}{2})\frac{{\partial }^l}{\partial {\alpha }^l}[K_{\alpha }(xz)] \end{gathered}$$

(66)

By (62), we have

$$\begin{gathered} {\displaystyle {\int }_0^1{(1-t^2)}^{\alpha -\frac{1}{2}}}\left\{\begin{array}{c}\sin (xt)\\ \sinh (xt)\end{array}\right\}{\ln }^m(1-t^2)dt \\ =\frac{\sqrt{\pi }{(\frac{2}{x})}^{\alpha }\mathrm{\Gamma }(\frac{1}{2}+\alpha )}{2}{\displaystyle \sum _{l=0}^m\left(\begin{array}{c}m\\ l\end{array}\right)}{P}_{m-l,\alpha }^{+}(\frac{2}{x},1,\frac{1}{2})\frac{{\partial }^l}{\partial {\alpha }^l}\left[\left\{\begin{array}{c}{\mathrm{H}}_{\mathsf{\alpha }}(x)\\ {\mathrm{L}}_{\mathsf{\alpha }}(x)\end{array}\right\}\right] \end{gathered}$$

(67)

By (48) and (50), we have

$$\frac{1}{\pi }{\displaystyle {\int }_0^{\pi }{\theta }^m\left\{\begin{array}{c}\cos (\alpha \theta -x\sin \theta +\frac{m\pi }{2})\\ \sin (\alpha \theta -x\sin \theta +\frac{m\pi }{2})\end{array}\right\}d\theta }=\frac{{\partial }^m}{\partial {\alpha }^m}\left\{\begin{array}{c}{\mathrm{J}}_{\mathsf{\alpha }}(x)\\ {\mathrm{E}}_{\mathsf{\alpha }}(x)\end{array}\right\}$$

(68)

By (63), we have

$$\begin{gathered} {\displaystyle {\int }_1^{\infty }\frac{e^{izt}}{{(t^2-1)}^{\alpha +\frac{1}{2}}}{\ln }^m\frac{1}{t^2-1}}dt \\ =\frac{i{(\frac{z}{2})}^{\alpha }\sqrt{\pi }\mathrm{\Gamma }(\frac{1}{2}-\alpha )}{2}{\displaystyle \sum _{l=0}^m\left(\begin{array}{c}m\\ l\end{array}\right)}{P}_{m-l,\alpha }^{+}(\frac{z}{2},-1,\frac{1}{2})\frac{{\partial }^l}{\partial {\alpha }^l}[H_{\alpha }^{(1)}(z)] \end{gathered}$$

(69)

In the following, we obtain the following comparison of the numerical results for (64)–(69).

As can be seen from Table 3, some integrals related to $Y_{\alpha }(x)$, $K_{\alpha }(x)$, ${\mathrm{J}}_{\mathsf{\alpha }}(x)$, ${\mathrm{E}}_{\mathsf{\alpha }}(x)$, ${\mathrm{H}}_{\mathsf{\alpha }}(x)$ and ${\mathrm{L}}_{\mathsf{\alpha }}(x)$ can be calculated by the right sides of (64)–(69). These algorithms can accelerate calculation speed and improve precision. Therefore, the algorithms given in this paper are fast and high precision.

Table 3. The comparison of the numerical results for (64)–(69).

Algorithm	α,x,m	Time16, Err	Time32, Err	Time48, Err	Value
(64)L	$-\frac{1}{3},\frac{3}{4},4$	0.078125,10−9	0.093750,10−18	0.156250,10−26	4.96945368037296 9002463996299⋯
(64)R	$-\frac{1}{3},\frac{3}{4},4$	0.000000,10−18	0.015625,10−38	0.000000,10−58
(65)L	$-\frac{1}{3},\frac{7}{4},4$	0.015625,10−8	0.078125,10−12	0.093750,10−12	−675.25026031399 638583019078⋯
(65)R	$-\frac{1}{3},\frac{7}{4},3$	0.000000,10−19	0.015625,10−39	0.000000,10−59
(66)L	$-\frac{1}{3},\frac{1}{3},4 z=\frac{1}{5}$	0.046875,10−11	0.062500,10−19	0.093750,10−27	−83.493834077193 62379381270575⋯
(66)R	$-\frac{1}{3},\frac{1}{3},4 z=\frac{1}{5}$	0.015625,10−17	0.000000,10−37	0.000000,10−57
(67)L	$-\frac{1}{3},\frac{23}{7},4 \mathit{for}{\mathrm{H}}_{\mathrm{a}}$	0.031250,10−7	0.078125,10−11	0.140625,10−12	−13392.783730964 647162808034589⋯
(67)R	$-\frac{1}{3},\frac{23}{7},4 \mathit{for}{\mathrm{H}}_{\mathrm{a}}$	0.000000,10−18	0.000000,10−38	0.000000,10−58
(67)L	$-\frac{1}{3},\frac{23}{7},4 \mathit{for}{\mathrm{L}}_{\mathrm{a}}$	0.015625,10−7	0.078125,10−11	0.109375,10−12	1245195.773481521 556850797968369⋯
(67)R	$-\frac{1}{3},\frac{23}{7},4 \mathit{for}{\mathrm{L}}_{\mathrm{a}}$	0.000000,10−20	0.000000,10−40	0.000000,10−60
(68)L	$-\frac{1}{3},\frac{7}{4},5\mathit{for}{\mathrm{J}}_{\mathrm{a}}$	0.015625,10−13	0.015625,10−32	0.031250,10−48	47.47582681649346 187731701168501⋯
(68)R	$-\frac{1}{3},\frac{7}{4},5\mathit{for}{\mathrm{J}}_{\mathrm{a}}$	0.015625,10−19	0.031250,10−39	0.031250,10−592
(68)L	$-\frac{1}{3},\frac{7}{4},5\mathit{for}{\mathrm{E}}_{\mathrm{a}}$	0.015625,10−16	0.015625,10−32	0.015625,10−48	−0.47429674849304 849671297559750⋯
(68)R	$-\frac{1}{3},\frac{7}{4},5\mathit{for}{\mathrm{E}}_{\mathrm{a}}$	0.015625,10−17	0.015625,10−37	0.031250,10−57
(69)L	$-\frac{1}{3},\frac{3}{7},3$	0.093750,10−12	0.203125,10−18	0.296875,10−26	33.096824127964⋯ +29.4406483104⋯i
(69)R	$-\frac{1}{3},\frac{3}{7},3$	0.015625,10−19	0.015625,10−39	0.000000,10−59

4. Conclusions

In this paper, a recursive algorithm for ${}_0\widehat{F}{{}_1}^{(m,n)}(\alpha ,\beta ,x)$ is established for $m,n=1,2\cdots .$ Therefore, a quick calculation $\frac{{\partial }^m}{\partial {\alpha }^m}\left\{\begin{array}{c}{J}_{\alpha }(x)\\ I_{\alpha }(x)\end{array}\right\}$ is obtained by (28). Based on the calculation of $\frac{{\partial }^m}{\partial {\alpha }^m}\left\{\begin{array}{c}{J}_{\alpha }(x)\\ I_{\alpha }(x)\end{array}\right\}$, we obtain the calculation of $\frac{{\partial }^m}{\partial {\alpha }^m}[Y_{\alpha }(x)]$, $\frac{{\partial }^m}{\partial {\alpha }^m}[K_{\alpha }(x)]$ and $\frac{{\partial }^m}{\partial {\alpha }^m}\left\{\begin{array}{c}{\mathrm{H}}_{\mathsf{\alpha }}(x)\\ {\mathrm{L}}_{\mathsf{\alpha }}(x)\end{array}\right\}$, $\frac{{\partial }^m}{\partial {\alpha }^m}\left\{\begin{array}{c}{\mathrm{J}}_{\mathsf{\alpha }}(x)\\ {\mathrm{E}}_{\mathsf{\alpha }}(x)\end{array}\right\}$. Furthermore, some integrals can be expressed in terms of derivatives of related Bessel functions with respect to the parameter. Numerical examples show the algorithms given in this paper are fast and high precision.