Successive differentiation of some mathematical functions using hypergeometric mechanism

In this article, we obtain successive differentiation of some composite mathematical functions: ${\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo>}}^{-\frac{1}{2}}{{\displaystyle \sin }}^{-1}\sqrt{\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo>}}+\sqrt{\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}$ ; ${\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo>}}^{\frac{1}{2}}{{\displaystyle \sin }}^{-1}\sqrt{\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo>}}+\sqrt{\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}$ ; $\frac{4}{z}\left[1-\sqrt{\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}+\ell n\left(\frac{1+\sqrt{\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}}{2}\right)\right]$ ; $\frac{4}{z^2}\left[2\sqrt{\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}-2+z-2z\ell n\left(\frac{1+\sqrt{\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}}{2}\right)\right]$ and $-\frac{4}{z}\ell n\left(\frac{1+\sqrt{\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}}{2}\right)$ , using a hypergeometric approach as the successive differentiation of these functions can not be performed by any other mathematical technique.

2020 MSC: 33C05, 33B10

1. Introduction and preliminaries

The ${}_p{F}_q\left(p\mathrm{,}q\in {\mathbb{N}}_0\right)$ is the generalized hypergeometric series defined by (see, e.g., [1-6]):

$\begin{array}{l}{}_p{F}_q\left[\begin{array}{l}{\alpha }_1\mathrm{,}\dots \mathrm{,}{\alpha }_p\mathrm{<mo>;</mo>}\\ {\beta }_1\mathrm{,}\dots \mathrm{,}{\beta }_q\mathrm{<mo>;</mo>}z\end{array}\right]\mathrm{<mo>=</mo>}{\displaystyle \sum _{n\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>}{\alpha }_1\mathrm{<mo\; stretchy="false">)</mo>}}_n\cdots {\mathrm{<mo\; stretchy="false">(</mo>}{\alpha }_p\mathrm{<mo\; stretchy="false">)</mo>}}_n}{{\mathrm{<mo\; stretchy="false">(</mo>}{\beta }_1\mathrm{<mo\; stretchy="false">)</mo>}}_n\cdots {\mathrm{<mo\; stretchy="false">(</mo>}{\beta }_q\mathrm{<mo\; stretchy="false">)</mo>}}_n}\frac{z^n}{n\mathrm{<mo>!</mo>}}\\ {=}_p{F}_q({\alpha }_1,\mathrm{...},{\alpha }_p;{\beta }_1\mathrm{,}\dots \mathrm{,}{\beta }_q\mathrm{<mo>;</mo>}z),\text{ (1}\text{.1)}\end{array}$

Being a natural generalization of the Gaussian hypergeometric series ${}_2{F}_1$ , where (λ)v denotes the Pochhammer symbol (for $\lambda \mathrm{,}\nu \in ℂ$ ) defined by

${\mathrm{<mo\; stretchy="false">(</mo>}\lambda \mathrm{<mo\; stretchy="false">)</mo>}}_{\nu }\mathrm{<mo>:</mo><mo>=</mo>}\frac{\Gamma \mathrm{<mo\; stretchy="false">(</mo>}\lambda +\nu \mathrm{<mo\; stretchy="false">)</mo>}}{\Gamma \mathrm{<mo\; stretchy="false">(</mo>}\lambda \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}\lambda \mathrm{,}\nu +\lambda \in ℂ\backslash {\mathbb{Z}}_0^{-}\mathrm{<mo\; stretchy="false">)</mo>}$

$\begin{array}{l}\left\{\begin{array}{l}1\\ \lambda \text{(}\lambda \text{+1)}\mathrm{...}\text{(}\lambda \text{+n-1}\end{array}\right\}(v=0;\lambda \in ℂ\backslash {\mathbb{Z}}_0^{-}),\\ (v=n\in \mathbb{N};\lambda \in ℂ\backslash ),\text{ (1}\text{.2)}\end{array}$

Here $\Gamma$ is the familiar Gamma function (see, e.g., [5, Section 1.1]), and it is assumed that (0)0 : = 1, an empty product as 1, and that the variable z, the numerator parameters α1, …, αp and the denominator parameters β1, ….,βq take on complex values, provided that no zero appear in the denominator of (1.1), that is, that

$\mathrm{<mo\; stretchy="false">(</mo>}{\beta }_j\in ℂ\backslash {\mathbb{Z}}_0^{-}\mathrm{<mo>;</mo>}j\mathrm{<mo>=</mo>1,}\dots \mathrm{,}q\mathrm{<mo\; stretchy="false">)</mo>.}$

Here $\mathbb{Z}\text{, }ℝ\text{ and }ℂ$ elsewhere, let , and be respectively the sets of integers, real numbers, and complex numbers, and let

$\mathbb{N}\mathrm{<mo>:</mo><mo>=</mo><mo>\{</mo>1,}\mathrm{2,}3\dots \mathrm{<mo>\}</mo><mo>;</mo>}{\mathbb{N}}_0\mathrm{<mo>:</mo><mo>=</mo>}\mathbb{N}\cup \left\{0\right\}\mathrm{<mo>;</mo>}{\mathbb{Z}}_0^{-}\mathrm{<mo>:</mo><mo>=</mo>}{\mathbb{Z}}^{-}\cup \left\{0\right\}\mathrm{<mo>=</mo>}\left\{\mathrm{0,}-\mathrm{1,}-\mathrm{2,}-\mathrm{3,}\cdots \right\}\mathrm{.}$

For more details of ${}_p{F}_q$ including its convergence, its various special and limiting cases, and its further diverse generalizations, one may refer to [7,8]. Certain identities associated with the ${}_p{F}_q$ and its generalizations, which are necessary for this work, are brought to mind.

See ref. [4, p.71, Q.No.(18)]

$\frac{{{\displaystyle \sin }}^{-1}\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo>}}{\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo>}}{\mathrm{<mo>=</mo>}}_2{F}_1\left[\begin{array}{ccc}\frac{1}{2}\mathrm{,<mi>\; </mi>}\frac{1}{2}\mathrm{<mo>;</mo>}& & \\ & z^2& \\ \mathrm{<mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi>}\frac{3}{2}\mathrm{<mo>;</mo>}& & \\ & & \end{array}\right]\mathrm{<mo>;</mo><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi>}\left|z\right|\mathrm{<mo><</mo>1.<mi>\; </mi><mi>\; </mi>}\text{ (1}\text{.3)}$

See ref. [6, p.44, Eq.(8)]

${\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}^{-a}{\mathrm{<mo>=</mo><mi>\; </mi>}}_1{F}_0\left[\begin{array}{cc}a\mathrm{<mi>\; </mi><mi>\; </mi><mo>;</mo>}& \\ & z\\ \mathrm{<mi>\; </mi>}-\mathrm{<mo>;</mo>}& \\ & \end{array}\right]\mathrm{<mo>;</mo><mi>\; </mi><mi>\; </mi><mi>\; </mi>}a\in ℂ\mathrm{<mi>\; </mi>}\text{and}\mathrm{<mi>\; </mi><mo>|</mo>}z\mathrm{<mo>|</mo><mo><</mo>1.<mi>\; </mi>}\text{ (1}\text{.4)}$

See ref. [9, p.155, Eq.(2.1)]

$-\frac{4}{z}\mathrm{<mi>\; </mi>}\ell n\left(\frac{1+\sqrt{1-z}}{2}\right){\mathrm{<mo>=</mo>}}_3{F}_2\left[\begin{array}{ccc}\mathrm{1,<mi>\; </mi>1,<mi>\; </mi>}\frac{3}{2}\mathrm{<mo>;</mo>}& & \\ & z& \\ \mathrm{<mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi>2,<mi>\; </mi>2<mo>;</mo>}& & \\ & & \end{array}\right]\mathrm{<mo>;</mo><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi>}\left|z\right|\mathrm{<mo><</mo>1.<mi>\; </mi>}\text{ (1}\text{.5)}$

Whenever the generalized hypergeometric function ${}_p{F}_q$ , including ${}_2{F}_1$ , can be expressed in terms of Gamma functions through summation of its specified argument, which may include unit or argument, the outcome holds significant value from both theoretical and practical perspectives.

The generalized hypergeometric series has classical summation theorems, including those of Gauss, Gauss second, Kummer, and Bailey for the ${}_2{F}_1$ series, as well as Watson's, Dixon's, Whipple's, and Saalschütz's summation theorems for the ${}_3{F}_2$ series and others. These theorems have significant importance in both theory and application.

From 1992 to 1996, Lavoie et al. [10-12] published a series of works that generalized the aforementioned classical summation theorems for the ${}_3{F}_2$ series of Watson, Dixon, and Whipple. They also presented many special and limiting cases of their results, which have been further extended and generalized by Rakha-Rathie [13], Kim et al. [14], and more recently by Qureshi et al. [15]. These results have also been verified, using computer programs such as Mathematica.

The emergence of extensively generalized special functions, such as (1.1), has sparked intriguing research into their reducibility. Bhat et al. introduce certain hypergeometric functions involving arcsine (x) using the Maclaurin series and their applications [16]. Qureshi et al. [17] also introduced hypergeometric forms of some composite functions containing arccosine (x) using the same series. Many papers from Qureshi et al.[18,19] introduced hypergeometric forms of some functions involving arcsine (x) using a differential equation approach and some mathematical functions via the Maclaurin series.

Inspired by the aforementioned papers, especially [9] comparing the resulting ordinary differential equations with standard ordinary differential equations of Leibnitz and Gauss, obtained some new hypergeometric functions. Our objective is to introduce successive differentiation of some composite functions by using a hypergeometric approach. For that we mention the hypergeometric forms of some composite functions in section 2, with their proof in section 3, using the series rearrangement technique. Applications of these hypergeometric forms in successive differentiation (mentioned in section 4), are given in section 5.

2. Hypergeometric forms of some mathematical functions

When |z|<1, the following hypergeometric forms of mathematical functions hold true:

$\frac{4}{z^2}\left[2\sqrt{\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}-2+z-2z\ell n\left(\frac{1+\sqrt{\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}}{2}\right)\right]{\mathrm{<mo>=</mo>}}_3{F}_2\left[\begin{array}{ccc}\mathrm{1,<mi>\; </mi>1,<mi>\; </mi>}\frac{3}{2}\mathrm{<mo>;</mo>}& & \\ & z& \\ \mathrm{<mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi>2,<mi>\; </mi>3<mo>;</mo>}& & \\ & & \end{array}\right]\mathrm{.}\text{ (2}\text{.1)}$

${\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo>}}^{\frac{1}{2}}{{\displaystyle \sin }}^{-1}\sqrt{\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo>}}+\sqrt{\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}{\mathrm{<mo>=</mo>}}_2{F}_1\left[\begin{array}{ccc}\frac{-1}{2}\mathrm{,<mi>\; </mi>}\frac{-1}{2}\mathrm{<mo>;</mo>}& & \\ & z& \\ \mathrm{<mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi>}\frac{1}{2}\mathrm{<mo>;</mo>}& & \\ & & \end{array}\right]\mathrm{.<mi>\; </mi>}\text{ (2}\text{.2)}$

$\frac{4}{z}\left[1-\sqrt{\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}+\ell n\left(\frac{1+\sqrt{\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}}{2}\right)\right]{\mathrm{<mo>=</mo>}}_3{F}_2\left[\begin{array}{ccc}\mathrm{1,<mi>\; </mi>1,<mi>\; </mi>}\frac{1}{2}\mathrm{<mo>;</mo>}& & \\ & z& \\ \mathrm{<mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi>2,<mi>\; </mi>2<mo>;</mo>}& & \\ & & \end{array}\right]\mathrm{.}\text{ (2}\text{.3)}$

${\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo>}}^{-\frac{1}{2}}{{\displaystyle \sin }}^{-1}\sqrt{\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo>}}+\sqrt{\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}{\mathrm{<mo>=</mo>2.}}_2{F}_1\left[\begin{array}{ccc}\frac{-1}{2}\mathrm{,<mi>\; </mi>}\frac{1}{2}\mathrm{<mo>;</mo>}& & \\ & z& \\ \mathrm{<mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi>}\frac{3}{2}\mathrm{<mo>;</mo>}& & \\ & & \end{array}\right]\mathrm{.<mi>\; </mi>}\text{ (2}\text{.4)}$

3. Derivation of hypergeometric forms

In this section, using the series rearrangement technique, we derive the hypergeometric forms of some mathematical functions mentioned in section 2.

Proof of hypergeometric form (2.1)

$\text{Let}\mathrm{<mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi>}{H}_1\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}\frac{4}{z^2}\left[2\sqrt{\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}-2+z-2z\ell n\left(\frac{1+\sqrt{\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}}{2}\right)\right]$

$\mathrm{<mo>=</mo>}\frac{\mathrm{8<mo\; stretchy="false">(</mo>1}-z{\mathrm{<mo\; stretchy="false">)</mo>}}^{\frac{1}{2}}}{z^2}-\frac{8}{z^2}+\frac{4}{z}-\frac{8}{z}\ell n\left(\frac{1+\sqrt{\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}}{2}\right)\mathrm{.}$

Using the equations (1.4) and (1.5), we have

$H_1\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}\frac{8}{z^2}{\mathrm{<mi>\; </mi>}}_1{F}_0\left[\begin{array}{ccc}\frac{-1}{2}\mathrm{<mo>;</mo>}& & \\ & z& \\ \mathrm{<mi>\; </mi>}-\mathrm{<mo>;</mo>}& & \\ & & \end{array}\right]-\frac{8}{z^2}+\frac{4}{z}+{\mathrm{2<mi>\; </mi>}}_3{F}_2\left[\begin{array}{ccc}\mathrm{1,<mi>\; </mi>1,<mi>\; </mi>}\frac{3}{2}\mathrm{<mo>;</mo>}& & \\ & z& \\ \mathrm{<mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi>2,<mi>\; </mi>2<mo>;</mo>}& & \\ & & \end{array}\right]\mathrm{<mi>\; </mi>}$

$\mathrm{<mo>=</mo>}\frac{8}{z^2}{\displaystyle \sum _{n\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>}\frac{-1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_n{z}^n}{n\mathrm{<mo>!</mo>}}-\frac{8}{z^2}+\frac{4}{z}+{\mathrm{2<mi>\; </mi>}}_3{F}_2\left[\begin{array}{ccc}\mathrm{1,<mi>\; </mi>1,<mi>\; </mi>}\frac{3}{2}\mathrm{<mo>;</mo>}& & \\ & z& \\ \mathrm{<mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi>2,<mi>\; </mi>2<mo>;</mo>}& & \\ & & \end{array}\right]\mathrm{<mi>\; </mi>}$

$\mathrm{<mo>=</mo>}\frac{8}{z^2}\left[1-\frac{z}{2}+{\displaystyle \sum _{n\mathrm{<mo>=</mo>2}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>}\frac{-1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_n{z}^n}{n\mathrm{<mo>!</mo>}}\right]-\frac{8}{z^2}+\frac{4}{z}+{\mathrm{2<mi>\; </mi>}}_3{F}_2\left[\begin{array}{ccc}\mathrm{1,<mi>\; </mi>1,<mi>\; </mi>}\frac{3}{2}\mathrm{<mo>;</mo>}& & \\ & z& \\ \mathrm{<mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi>2,<mi>\; </mi>2<mo>;</mo>}& & \\ & & \end{array}\right]$

$\mathrm{<mo>=</mo>}\frac{8}{z^2}{\displaystyle \sum _{n\mathrm{<mo>=</mo>2}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>}\frac{-1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_n{z}^n}{n\mathrm{<mo>!</mo>}}+{\mathrm{2<mi>\; </mi>}}_3{F}_2\left[\begin{array}{ccc}\mathrm{1,<mi>\; </mi>1,<mi>\; </mi>}\frac{3}{2}\mathrm{<mo>;</mo>}& & \\ & z& \\ \mathrm{<mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi>2,<mi>\; </mi>2<mo>;</mo>}& & \\ & & \end{array}\right]\mathrm{.<mi>\; </mi>}\text{ (3}\text{.1)}$

Replacing n by (n+2) in equation (3.1), we get

$H_1\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}\frac{8}{z^2}{\displaystyle \sum _{n\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>}\frac{-1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_{n+2}\mathrm{<mi>\; </mi>}{z}^{n+2}}{\mathrm{<mo\; stretchy="false">(</mo>}n+\mathrm{2<mo\; stretchy="false">)</mo><mo>!</mo>}}+{\mathrm{2<mi>\; </mi>}}_3{F}_2\left[\begin{array}{ccc}\mathrm{1,<mi>\; </mi>1,<mi>\; </mi>}\frac{3}{2}\mathrm{<mo>;</mo>}& & \\ & z& \\ \mathrm{<mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi>2,<mi>\; </mi>2<mo>;</mo>}& & \\ & & \end{array}\right]\mathrm{<mi>\; </mi><mi>\; </mi>}$

$\mathrm{<mo>=</mo>}\frac{8}{z^2}\frac{{\mathrm{<mo\; stretchy="false">(</mo>}\frac{-1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_2\mathrm{<mi>\; </mi>}{z}^2}{{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_2}{\displaystyle \sum _{n\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>}\frac{3}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_n\mathrm{<mi>\; </mi>}{z}^n}{{\mathrm{<mo\; stretchy="false">(</mo>3<mo\; stretchy="false">)</mo>}}_n}+{\mathrm{2<mi>\; </mi>}}_3{F}_2\left[\begin{array}{ccc}\mathrm{1,<mi>\; </mi>1,<mi>\; </mi>}\frac{3}{2}\mathrm{<mo>;</mo>}& & \\ & z& \\ \mathrm{<mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi>2,<mi>\; </mi>2<mo>;</mo>}& & \\ & & \end{array}\right]\mathrm{<mi>\; </mi>}$

$\mathrm{<mo>=</mo>4}\left(\frac{-1}{2}\right)\left(\frac{1}{2}\right){\displaystyle \sum _{n\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>}\frac{3}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_n\mathrm{<mi>\; </mi>}{z}^n{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_n}{{\mathrm{<mo\; stretchy="false">(</mo>3<mo\; stretchy="false">)</mo>}}_{n}n\mathrm{<mo>!</mo>}}+{\mathrm{2<mi>\; </mi>}}_3{F}_2\left[\begin{array}{ccc}\mathrm{1,<mi>\; </mi>1,<mi>\; </mi>}\frac{3}{2}\mathrm{<mo>;</mo>}& & \\ & z& \\ \mathrm{<mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi>2,<mi>\; </mi>2<mo>;</mo>}& & \\ & & \end{array}\right]$

$\mathrm{<mo>=</mo>}-{\displaystyle \sum _{n\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>}\frac{3}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_n\mathrm{<mi>\; </mi>}{z}^n}{{\mathrm{<mo\; stretchy="false">(</mo>3<mo\; stretchy="false">)</mo>}}_{n}n\mathrm{<mo>!</mo>}}+2{\displaystyle \sum _{n\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>}\frac{3}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_n{z}^n}{{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_{n}n\mathrm{<mo>!</mo>}}\mathrm{<mi>\; </mi>}$

$\mathrm{<mo>=</mo>2}{\displaystyle \sum _{n\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>}\frac{3}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_n{z}^n}{{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_{n}n\mathrm{<mo>!</mo>}}-{\displaystyle \sum _{n\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>}\frac{3}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_n\mathrm{<mi>\; </mi>}{z}^n}{{\mathrm{<mo\; stretchy="false">(</mo>3<mo\; stretchy="false">)</mo>}}_{n}n\mathrm{<mo>!</mo>}}\mathrm{<mi>\; </mi><mi>\; </mi>}$

$\mathrm{<mo>=</mo>}{\displaystyle \sum _{n\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>}\frac{3}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_n{z}^n}{n\mathrm{<mo>!</mo>}}\left[\frac{{\mathrm{2<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_n}{{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_n}-\frac{1}{{\mathrm{<mo\; stretchy="false">(</mo>3<mo\; stretchy="false">)</mo>}}_n}\right]\mathrm{<mi>\; </mi>}$

$\mathrm{<mo>=</mo>}{\displaystyle \sum _{n\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>}\frac{3}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_n{z}^n}{n\mathrm{<mo>!</mo>}}\left[\frac{2\Gamma \mathrm{<mo\; stretchy="false">(</mo>1}+n\mathrm{<mo\; stretchy="false">)</mo>}}{\Gamma \mathrm{<mo\; stretchy="false">(</mo>2}+n\mathrm{<mo\; stretchy="false">)</mo>}\Gamma \mathrm{<mo\; stretchy="false">(</mo>2}+n\mathrm{<mo\; stretchy="false">)</mo>}}-\frac{\Gamma \mathrm{<mo\; stretchy="false">(</mo>3<mo\; stretchy="false">)</mo>}}{\Gamma \mathrm{<mo\; stretchy="false">(</mo>3}+n\mathrm{<mo\; stretchy="false">)</mo>}}\right]$

$\mathrm{<mo>=</mo>}{\displaystyle \sum _{n\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>}\frac{3}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_n{z}^n}{n\mathrm{<mo>!</mo>}}\left[\frac{2\Gamma \mathrm{<mo\; stretchy="false">(</mo>1}+n\mathrm{<mo\; stretchy="false">)</mo>}}{\mathrm{<mo\; stretchy="false">(</mo>1}+n\mathrm{<mo\; stretchy="false">)</mo>}\Gamma \mathrm{<mo\; stretchy="false">(</mo>1}+n\mathrm{<mo\; stretchy="false">)</mo>}\Gamma \mathrm{<mo\; stretchy="false">(</mo>2}+n\mathrm{<mo\; stretchy="false">)</mo>}}-\frac{2}{\mathrm{<mo\; stretchy="false">(</mo>2}+n\mathrm{<mo\; stretchy="false">)</mo>}\Gamma \mathrm{<mo\; stretchy="false">(</mo>2}+n\mathrm{<mo\; stretchy="false">)</mo>}}\right]\mathrm{<mi>\; </mi>}$

$\mathrm{<mo>=</mo>}{\displaystyle \sum _{n\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>}\frac{3}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_n{z}^n}{n\mathrm{<mo>!</mo>}}\left[\frac{2}{\Gamma \mathrm{<mo\; stretchy="false">(</mo>2}+n\mathrm{<mo\; stretchy="false">)</mo>}}\left(\frac{1}{\mathrm{<mo\; stretchy="false">(</mo>1}+n\mathrm{<mo\; stretchy="false">)</mo>}}-\frac{1}{\mathrm{<mo\; stretchy="false">(</mo>2}+n\mathrm{<mo\; stretchy="false">)</mo>}}\right)\right]\mathrm{<mi>\; </mi>}$

$\mathrm{<mo>=</mo>}{\displaystyle \sum _{n\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>}\frac{3}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_n{z}^n}{n\mathrm{<mo>!</mo>}}\left[\frac{2}{{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_n\Gamma \mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}\left(\frac{1}{\mathrm{<mo\; stretchy="false">(</mo>1}+n\mathrm{<mo\; stretchy="false">)</mo>}}-\frac{1}{\mathrm{<mo\; stretchy="false">(</mo>2}+n\mathrm{<mo\; stretchy="false">)</mo>}}\right)\right]\mathrm{<mi>\; </mi>}$

$H_1\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{\displaystyle \sum _{n\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{2<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>}\frac{3}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_n{z}^n}{{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_n\Gamma \mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}n\mathrm{<mo>!</mo>}}\left[\frac{1}{\mathrm{<mo\; stretchy="false">(</mo>1}+n\mathrm{<mo\; stretchy="false">)</mo>}}-\frac{1}{\mathrm{<mo\; stretchy="false">(</mo>2}+n\mathrm{<mo\; stretchy="false">)</mo>}}\right]\mathrm{.<mi>\; </mi>}$

After further simplification, we get the result (2.1).

Proof of hypergeometric form (2.2)

$\text{Let}\mathrm{<mi>\; </mi><mi>\; </mi>}{H}_2\mathrm{<mo\; stretchy="false">(</mo>}z{\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}}_2{F}_1\left[\begin{array}{ccc}\frac{-1}{2}\mathrm{,<mi>\; </mi>}\frac{-1}{2}\mathrm{<mo>;</mo>}& & \\ & z& \\ \mathrm{<mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi>}\frac{1}{2}\mathrm{<mo>;</mo>}& & \\ & & \end{array}\right]-\sqrt{\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}$

${\mathrm{<mo>=</mo>}}_2{F}_1\left[\begin{array}{ccc}\frac{-1}{2}\mathrm{,<mi>\; </mi>}\frac{-1}{2}\mathrm{<mo>;</mo>}& & \\ & z& \\ \mathrm{<mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi>}\frac{1}{2}\mathrm{<mo>;</mo>}& & \\ & & \end{array}\right]-{\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}^{\frac{1}{2}}\mathrm{.<mi>\; </mi>}$

Using the equation (1.4), we have

$H_2\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{\displaystyle \sum _{r\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>}-\frac{1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_r{\mathrm{<mo\; stretchy="false">(</mo>}-\frac{1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_r{z}^r}{{\mathrm{<mo\; stretchy="false">(</mo>}\frac{1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_r\mathrm{<mi>\; </mi>}r\mathrm{<mo>!</mo>}}{-}_1{F}_0\left[\begin{array}{ccc}\frac{-1}{2}\mathrm{<mo>;</mo>}& & \\ & z& \\ -\mathrm{<mo>;</mo>}& & \\ & & \end{array}\right]\mathrm{<mi>\; </mi><mi>\; </mi>}$

$\mathrm{<mo>=</mo>}{\displaystyle \sum _{r\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>}-\frac{1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_r{\mathrm{<mo\; stretchy="false">(</mo>}-\frac{1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_r{z}^r}{{\mathrm{<mo\; stretchy="false">(</mo>}\frac{1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_r\mathrm{<mi>\; </mi>}r\mathrm{<mo>!</mo>}}-{\displaystyle \sum _{r\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>}-\frac{1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_r{z}^r}{r\mathrm{<mo>!</mo>}}$

$\mathrm{<mo>=</mo>}{\displaystyle \sum _{r\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>}-\frac{1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_r{z}^r}{r\mathrm{<mo>!</mo>}}\left[\frac{{\mathrm{<mo\; stretchy="false">(</mo>}-\frac{1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_r}{{\mathrm{<mo\; stretchy="false">(</mo>}\frac{1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_r}-1\right]\mathrm{<mi>\; </mi>}$

$\mathrm{<mo>=</mo>}{\displaystyle \sum _{r\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>}-\frac{1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_r{z}^r}{r\mathrm{<mo>!</mo>}}\left[\frac{\Gamma \mathrm{<mo\; stretchy="false">(</mo>}-\frac{1}{2}+r\mathrm{<mo\; stretchy="false">)</mo>}\Gamma \mathrm{<mo\; stretchy="false">(</mo>}\frac{1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}{\Gamma \mathrm{<mo\; stretchy="false">(</mo>}-\frac{1}{2}\mathrm{<mo\; stretchy="false">)</mo>}\Gamma \mathrm{<mo\; stretchy="false">(</mo>}\frac{1}{2}+r\mathrm{<mo\; stretchy="false">)</mo>}}-1\right]\mathrm{<mi>\; </mi>}$

$\mathrm{<mo>=</mo>}{\displaystyle \sum _{r\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>}-\frac{1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_r{z}^r}{r\mathrm{<mo>!</mo>}}\left[\frac{\Gamma \mathrm{<mo\; stretchy="false">(</mo>}-\frac{1}{2}+r\mathrm{<mo\; stretchy="false">)</mo>}\sqrt{\pi }}{\mathrm{<mo\; stretchy="false">(</mo>}-2\sqrt{\pi }\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">(</mo>}-\frac{1}{2}+r\mathrm{<mo\; stretchy="false">)</mo>}\Gamma \mathrm{<mo\; stretchy="false">(</mo>}-\frac{1}{2}+r\mathrm{<mo\; stretchy="false">)</mo>}}-1\right]\mathrm{<mi>\; </mi>}$

$\mathrm{<mo>=</mo>}{\displaystyle \sum _{r\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>}-\frac{1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_r{z}^r}{r\mathrm{<mo>!</mo>}}\left[\frac{1}{1-2r}-1\right]\mathrm{<mi>\; </mi>}$

$\mathrm{<mo>=</mo>}{\displaystyle \sum _{r\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>}-\frac{1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_r{z}^r}{r\mathrm{<mo>!</mo>}}\left[\frac{2r}{1-2r}\right]\mathrm{<mi>\; </mi>}$

$H_2\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>2}{\displaystyle \sum _{r\mathrm{<mo>=</mo>1}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>}-\frac{1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_r\mathrm{<mi>\; </mi>}{z}^r}{\mathrm{<mo\; stretchy="false">(</mo>}r-\mathrm{1<mo\; stretchy="false">)</mo><mo>!</mo><mo\; stretchy="false">(</mo>1}-2r\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{.<mi>\; </mi>}\text{ (3}\text{.2)}$

Replacing r by (r+1) in equation (3.2), we get

$\mathrm{<mo>=</mo>2}{\displaystyle \sum _{r\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>}-\frac{1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_{r+1}\mathrm{<mi>\; </mi>}{z}^{r+1}}{\mathrm{<mo\; stretchy="false">(</mo>}r+1-\mathrm{1<mo\; stretchy="false">)</mo><mo>!</mo><mo\; stretchy="false">(</mo>1}-2r-\mathrm{2<mo\; stretchy="false">)</mo>}}\mathrm{<mi>\; </mi><mi>\; </mi>}$

$H_2\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}\frac{\mathrm{2<mo\; stretchy="false">(</mo>}-\frac{1}{2}\mathrm{<mo\; stretchy="false">)</mo><mi>\; </mi>}z}{\mathrm{<mo\; stretchy="false">(</mo>}-\mathrm{1<mo\; stretchy="false">)</mo>}}{\displaystyle \sum _{r\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>}-\frac{1}{2}+\mathrm{1<mo\; stretchy="false">)</mo>}}_r\mathrm{<mi>\; </mi>}{z}^r}{r\mathrm{<mo>!</mo><mo\; stretchy="false">(</mo>1}+2r\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{.<mi>\; </mi>}$

After further simplification, we get the result (2.2).

Similarly, we can get the remaining hypergeometric forms (2.3) and (2.4) in the same way as the hypergeometric forms (2.1) and (2.2).

4. Successive differential coefficients of some mathematical functions

When |z|<1, successive differential coefficients of some mathematical functions hold true:

$\frac{d^n}{d{z}^n}\left[-\frac{4}{z}\ell n\left(\frac{1+\sqrt{\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}}{2}\right)\right]\mathrm{<mo>=</mo>}{\frac{{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>}\frac{3}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_n}{{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_n}}_3{F}_2\left[\begin{array}{ccc}1+n\mathrm{,1}+n\mathrm{,}\frac{3}{2}+n\mathrm{<mo>;</mo>}& & \\ & z& \\ \mathrm{<mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi>2}+n\mathrm{,2}+n\mathrm{<mo>;</mo>}& & \\ & & \end{array}\right]\mathrm{.}\text{ (4}\text{.1)}$

$\begin{array}{l}\frac{d^n}{d{z}^n}\left[\frac{4}{z}\left\{1-\sqrt{\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}+\ell n\left(\frac{1+\sqrt{\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}}{2}\right)\right\}\right]\mathrm{<mo>=</mo>}\frac{{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>}\frac{1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_n}{{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_n}\times \mathrm{<mi>\; </mi>}\text{ (4}\text{.2)}\\ {\times }_3{F}_2\left[\begin{array}{ccc}1+n\mathrm{,<mi>\; </mi>1}+n\mathrm{,<mi>\; </mi>}\frac{1}{2}+n\mathrm{<mo>;</mo>}& & \\ & z& \\ \mathrm{<mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi>2}+n\mathrm{,<mi>\; </mi>2}+n\mathrm{<mo>;</mo>}& & \\ & & \end{array}\right]\mathrm{.}\end{array}$

$\begin{array}{l}\frac{d^n}{d{z}^n}\left[\frac{4}{z^2}\left\{2\sqrt{\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}-2+z-2z\ell n\left(\frac{1+\sqrt{\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}}{2}\right)\right\}\right]\mathrm{<mo>=</mo>}\frac{{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>}\frac{3}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_n}{{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>3<mo\; stretchy="false">)</mo>}}_n}\times \mathrm{<mi>\; </mi><mi>\; </mi>}\text{ (4}\text{.3)}\\ {\times }_3{F}_2\left[\begin{array}{ccc}1+n\mathrm{,<mi>\; </mi>1}+n\mathrm{,<mi>\; </mi>}\frac{3}{2}+n\mathrm{<mo>;</mo>}& & \\ & z& \\ \mathrm{<mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi>2}+n\mathrm{,<mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi>3}+n\mathrm{<mo>;</mo>}& & \\ & & \end{array}\right]\mathrm{.}\end{array}$

$\frac{d^n}{d{z}^n}\left[\frac{{{\displaystyle \sin }}^{-1}\sqrt{\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo>}}}{\sqrt{\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo>}}}+\sqrt{\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}\right]\mathrm{<mo>=</mo>}{\frac{\mathrm{2.<mo\; stretchy="false">(</mo>}-\frac{1}{2}{\mathrm{<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>}\frac{1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_n}{{\mathrm{<mo\; stretchy="false">(</mo>}\frac{3}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_n}}_2{F}_1\left[\begin{array}{ccc}-\frac{1}{2}+n\mathrm{,<mi>\; </mi>}\frac{1}{2}+n\mathrm{<mo>;</mo>}& & \\ & z& \\ \mathrm{<mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi>}\frac{3}{2}+n\mathrm{<mo>;</mo>}& & \\ & & \end{array}\right]\mathrm{.<mi>\; </mi>}\text{ (4}\text{.4)}$

$\frac{d^n}{d{z}^n}\left[\sqrt{\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo>}}{{\displaystyle \sin }}^{-1}\sqrt{\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo>}}+\sqrt{\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}\right]\mathrm{<mo>=</mo>}{\frac{{\mathrm{<mo\; stretchy="false">(</mo>}-\frac{1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>}-\frac{1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_n}{{\mathrm{<mo\; stretchy="false">(</mo>}\frac{1}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_n}}_2{F}_1\left[\begin{array}{ccc}-\frac{1}{2}+n\mathrm{,}-\frac{1}{2}+n\mathrm{<mo>;</mo>}& & \\ & z& \\ \mathrm{<mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi>}\frac{1}{2}+n\mathrm{<mo>;</mo>}& & \\ & & \end{array}\right]\mathrm{.<mi>\; </mi>}\text{ (4}\text{.5)}$

5. Applications of hypergeometric forms in successive differentiation

In this section, using the series rearrangement technique, we give the proof of successive differential coefficients of mathematical functions mentioned in Section 4.

Proof of successive differential coefficient (4.1)

$\text{Let}\mathrm{<mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi>}{D}_1\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}\frac{d^n}{d{z}^n}\left[-\frac{4}{z}\ell n\left(\frac{1+\sqrt{\mathrm{<mo\; stretchy="false">(</mo>1}-z\mathrm{<mo\; stretchy="false">)</mo>}}}{2}\right)\right]\mathrm{.<mi>\; </mi><mi>\; </mi>}$

Using the equation (1.5), we have

$D_1\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}\frac{d^n}{d{z}^n}\left({}_3{F}_2\left[\begin{array}{ccc}\mathrm{1,<mi>\; </mi>1,<mi>\; </mi>}\frac{3}{2}\mathrm{<mo>;</mo>}& & \\ & z& \\ \mathrm{<mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi><mi>\; </mi>2,<mi>\; </mi>2<mo>;</mo>}& & \\ & & \end{array}\right]\right)\mathrm{<mi>\; </mi>}$

$\mathrm{<mo>=</mo>}\frac{d^n}{d{z}^n}\left({\displaystyle \sum _{r\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_r{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_r{\mathrm{<mo\; stretchy="false">(</mo>}\frac{3}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_r\mathrm{<mi>\; </mi>}{z}^r}{{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_r{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_r\mathrm{<mi>\; </mi>}r\mathrm{<mo>!</mo>}}\right)\mathrm{<mi>\; </mi>}$

$\mathrm{<mo>=</mo>}\frac{d^n}{d{z}^n}\left({\displaystyle \sum _{r\mathrm{<mo>=</mo>0}}^{n-1}}\frac{{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_r{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_r{\mathrm{<mo\; stretchy="false">(</mo>}\frac{3}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_r\mathrm{<mi>\; </mi>}{z}^r}{{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_r{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_r\mathrm{<mi>\; </mi>}r\mathrm{<mo>!</mo>}}+{\displaystyle \sum _{r\mathrm{<mo>=</mo>}n}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_r{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_r{\mathrm{<mo\; stretchy="false">(</mo>}\frac{3}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_r\mathrm{<mi>\; </mi>}{z}^r}{{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_r{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_r\mathrm{<mi>\; </mi>}r\mathrm{<mo>!</mo>}}\right)\mathrm{<mi>\; </mi>}$

$\mathrm{<mo>=</mo>0}+{\displaystyle \sum _{r\mathrm{<mo>=</mo>}n}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_r{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_r{\mathrm{<mo\; stretchy="false">(</mo>}\frac{3}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_r\mathrm{<mi>\; </mi>}}{{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_r{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_r\mathrm{<mi>\; </mi>}r\mathrm{<mo>!</mo>}}\frac{d^n}{d{z}^n}{z}^r$

$D_1\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{\displaystyle \sum _{r\mathrm{<mo>=</mo>}n}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_r{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_r{\mathrm{<mo\; stretchy="false">(</mo>}\frac{3}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_r\mathrm{<mi>\; </mi>}{z}^{\mathrm{<mo\; stretchy="false">(</mo>}r-n\mathrm{<mo\; stretchy="false">)</mo>}}r\mathrm{<mo>!</mo>}}{{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_r{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_r\mathrm{<mi>\; </mi><mo\; stretchy="false">(</mo>}r-n\mathrm{<mo\; stretchy="false">)</mo><mo>!</mo>}r\mathrm{<mo>!</mo>}}\mathrm{.<mi>\; </mi>}\text{ (5}\text{.1)}$

Replacing r by (r+n) in equation (5.1), we get

$D_1\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{\displaystyle \sum _{r\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_{r+n}{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_{r+n}{\mathrm{<mo\; stretchy="false">(</mo>}\frac{3}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_{r+n}\mathrm{<mi>\; </mi>}{z}^{\mathrm{<mo\; stretchy="false">(</mo>}r+n-n\mathrm{<mo\; stretchy="false">)</mo>}}}{{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_{r+n}{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_{r+n}\mathrm{<mi>\; </mi><mo\; stretchy="false">(</mo>}r-n+n\mathrm{<mo\; stretchy="false">)</mo><mo>!</mo>}}$

$\mathrm{<mo>=</mo>}\frac{{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>}\frac{3}{2}\mathrm{<mo\; stretchy="false">)</mo>}}_n}{{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_n{\mathrm{<mo\; stretchy="false">(</mo>2<mo\; stretchy="false">)</mo>}}_n}{\displaystyle \sum _{r\mathrm{<mo>=</mo>0}}^{\infty }}\frac{{\mathrm{<mo\; stretchy="false">(</mo>1}+n\mathrm{<mo\; stretchy="false">)</mo>}}_r{\mathrm{<mo\; stretchy="false">(</mo>1}+n\mathrm{<mo\; stretchy="false">)</mo>}}_r{\mathrm{<mo\; stretchy="false">(</mo>}\frac{3}{2}+n\mathrm{<mo\; stretchy="false">)</mo>}}_r\mathrm{<mi>\; </mi>}{z}^r}{{\mathrm{<mo\; stretchy="false">(</mo>2}+n\mathrm{<mo\; stretchy="false">)</mo>}}_r{\mathrm{<mo\; stretchy="false">(</mo>2}+n\mathrm{<mo\; stretchy="false">)</mo>}}_r\mathrm{<mi>\; </mi>}r\mathrm{<mo>!</mo>}}\mathrm{.<mi>\; </mi>}$

After further simplification, we get the result (4.1).

Similarly, we can get the remaining successive differential coefficients (4.2)-(4.5) in the same way as the successive differential coefficient (4.1).

Conclusion

In this paper, we have obtained the hypergeometric forms of some composite functions. Further, we have found some applications of these hypergeometric forms in successive differentiation. We conclude our present investigation with the remark that the successive differentiation of some other functions can be derived in an analogous manner. Moreover, the results deduced above are quite significant and are expected to lead to some potential applications in several diverse fields of mathematical, physical, statistical, and engineering sciences.

The authors wish to extend their heartfelt appreciation to the anonymous reviewers for their invaluable feedback. Their constructive and encouraging comments have greatly contributed to enhancing the quality of this paper.

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