Integral transforms involving the product of Humbert and Bessel functions and its application

1.   Introduction

Integral transforms involving special functions have gained considerable attention in the bibliography. In the last decade, many papers are investigated to study the integral transforms involving the product of Bessel and other special functions (see ^[4,8,9]). In view of this, it is worth investigating a general integral transform involving Humbert and Bessel functions with a weight $e^{-\gamma x^{2}}$. This integral transform is important in evaluating the expression of the generalized Humbert-Gaussian beams propagating in the space. A closed form of the considered integral will be derived. To the best of our knowledge, the results of the present contribution have not been previously published.

The following definitions are essential to recall for the present investigation:

The four two-variables hypergeometric functions, called confluent Humbert functions ^[7], are defined by

and

with $|x| < 1, |y| < \infty\; \text{and}\; c, d\neq 0, -1, -2, \ldots$

The hypergeometric function ${_2}F_{1}$ is defined by (see ^[1])

where $(\alpha)_{n}$ is the Pochhammer symbol defined by

with $\Gamma$ is the gamma function (see ^[1]).

The Kummer function is defined by the series

In terms of the hypergeometric function (1.5), it's easy to obtain Humbert functions (1.1)–(1.4) (see ^[13]). Therefore, we can write

and

with $|x| < 1, |y| < \infty\; \text{and}\; c, d\neq 0, -1, -2, \ldots$

The four Appell series ^[2], which are double hypergeometric series, are defined by

with $\max\lbrace|x|, |y|\rbrace < 1$;

with $|x|+|y| < 1$;

with $\max\lbrace|x|, |y|\rbrace < 1$;

and

with $\sqrt{|x|}+\sqrt{|y|} < 1$.

These expressions, with c and $c^{\prime}$ are neither zero nor a negative integer, can be expressed in terms of ${_2}F{_1}$ as follows (see ^[13])

and

2.   Main results

2.1.   Evaluation of the integral $I_{m}$

In this section, we now evaluate the following integral transform containing Humbert functions $\; \text{ $\mathcal{F}$ }\; ( = \Psi_{1}, \Phi_{1}, \Xi_{1} \; \text{or}\; \Xi_{2})$:

In (2.1), we replaced $x$ and $y$ with $y$ and $xt^{2}$ respectively in (1.1), (1.2), (1.3) and (1.4).

Theorem 2.1. The following transformations hold true:

and

where, $\mathcal{A} = \frac{1}{\beta\gamma^{m}}\left(\frac{\beta}{2\gamma}\right) ^{d}(d)_{m}$.

Proof. The proof of integral transform in (2.2) is as follows:

By substituting (1.7) in (2.1), we obtain

where

with

By means of the identity ^[6]

and using (1.6), (2.7) can be written as

Substituting (2.10) into (2.6), the assertion (2.1) of Theorem 2.1 directly follows.

Note that if $m = 0 \; \text{and}\; c = d$, one finds the undermentioned integral:

Proof. The proof of integral transform (2.3) is as follows:

Considering $\mathcal{F}(y, xt^{2}) = \Phi_{1}\left(a, b; d; y, xt^{2}\right)$ given by (1.8), (2.1) becomes in this case

where $I_{q}$ is given by (2.7).

By using the identity (2.9), (2.12) can be written as

Substituting this last equation in the expression of $I_{m}(\Phi_{1})$, yields the assertion (2.3) of Theorem 2.1.

Note here that for $m = 0$ and $c = d$, (2.12) becomes

Proof. The proof of the integral transform (2.4) is as follows:

Following the same procedure as above, we use the definition of $\Xi_{1}$ given by (1.9) and replace $\mathcal{F}(y, xt^{2})$ by $\mathcal{F}(y, xt^{2}) = \Xi_{1}\left(a, b, c; c^{\prime}; y, xt^{2}\right)$, then the considered integral $I_{m}(\Xi_{1})$ can be written as

where $I_{q}$ is given by (2.7).

Now, by using (2.9) and substituting (2.7) in (2.15), (2.4) is proved.

By putting $m = 0 \; \text{and}\; c = d$, (2.15) can be written as

Proof. The proof of the integral transform (2.5) is as follows:

Taking $\mathcal{F}(y, xt^{2}) = \Xi_{2}\left(a, b; d; y, xt^{2}\right)$ given by (1.10), (2.1) becomes

Finally, by introducing the expression of $I_{q}$ given by (2.10), one finds (2.5). This completes the proof of Theorem 2.1.

For $m = 0 \; \text{and}\; c = d$, (2.17) becomes

2.2.   Evaluation of some series

In this section, we evaluate the series obtained from the integral transforms given by Theorem 2.1 in the case of $m = 0 \; \text{and}\; c = d$. These series are given by (see (2.11), (2.14), (2.16) and (2.18)).

and

Theorem 2.2. The following result holds true:

We note that in (2.24), $\mathcal{G} = \frac{e^{-\frac{Y}{X}}}{(1-X)^{-a}}$, $X = \frac{x}{x-1}$, $Y = \frac{xz}{1-x}$ and $Z = \frac{y}{1-x}$.

Proof of Eq. (2.23): We start from (2.19), which, by using the identities (see ^[10,12])

and

can be rearranged as

where

With the help of (1.5) and (2.27), this last expression can be written as

By using the Euler's transformation ^[11]

(2.29) becomes

where, the double summation in this last expression is $\Phi_{1}\left(a+n, c'-c; c'; \frac{x}{x-1}, \frac{xz}{1-x}\right)$ given by (1.2). Therefore, the result in (2.23) is proved.

Proof of Eq. (2.24): To prove (2.24), we start from (2.23) by puting

so that, the summation $\mathcal{S}_{1}$ can be written as

which, by using the identity

becomes

After some simplifications and by using (1.15), and the elementary identities

and

(2.37) can be expressed in terms of the Appell function $F_{2}$ as given in (2.24). This completes the proof.

Proof of Eq. (2.25): We start from the expression of $\mathcal{S}_{1}$ and with the help of the identities

$\mathcal{S}_{1}$ can be written in terms of ${_2}F_{1}$ as follows

If one uses (1.15), $\mathcal{S}_{1}$ can be written in terms of the second Appell function $F_{2}\left(a, c+n, b; c', c; x, y \right)$. This completes the proof of (2.25) and consequently Theorem 2.2.

Theorem 2.3. The following result holds true:

Proof of Eq. (2.42): To prove the result (2.42), we use the identities given by (2.26), (2.27) and (2.28) and the result ^[10]

to rewrite the expression of $\mathcal{S}_{2}$ given by (2.20). Consequently, it becomes

which can be written as

The summation in (2.47) can be rearranged, by using the Euler's transformation given by (2.32), as

Now, by employing the following identities

The expression of $\mathcal{S}_{2}$ can be written as

Since, the double summation in (2.50) is the Humbert function given by (1.2), that is, $\Phi_{1}\left(a+n, n; c+n; \frac{x}{x-1}, \frac{xz}{1-x} \right)$. Therefore, the required proof of (2.42) straightforwardly follows.

Proof of Eq. (2.43): The summation in (2.50) can be rewritten as

which after introducing the hypergeometric function, becomes

By using the expression of the second Appell function given by (1.15), (2.52) can be rearranged to yield (2.43). This completes the proof.

Proof of Eq.(2.44): By using the elementary identities

and

(2.20) can be written as

With the help of the definition of the first Appell function given by (2.55), we can easily obtain (2.44). This completes the proof of Theorem 2.3.

Theorem 2.4. The following result holds true:

Proof of Eq. (2.56): By the use of (2.28), (2.53) and (2.54), $\mathcal{S}_{3}$ can be expressed as

By replacing the last summation in (2.59) by the hypergeometric function ${_2}F_{1}$ and using the Euler's transformation given by (2.32), we find

The last summation in this equation is none other than the Humbert function $\Phi_{1}\left(b, c^{\prime}+ n-c; c^{\prime}+n; \frac{x}{x-1}, \frac{xz}{1-x}\right)$ given by (1.2). From here, the proof of (2.56) straightforwardly follows.

Proof of Eq. (2.57): We start from (2.59) which can be written, with the help of (2.32), as

Now, by recalling (1.14), $\mathcal{S}_{3}$ can be rewritten in terms of the first Appell function $F_{1}\left(a, b, b^{\prime}; c; x, y \right)$ as expressed in (2.57). This completes the proof.

Proof of Eq. (2.58): We write (2.21) of $\mathcal{S}_{3}$ as

which, with the help of the identities given by (2.27), becomes

Recalling the expression of the third Appell function $F_{3}$ given by (1.16) that yields (2.58) in terms of series of this function. This completes the proof of (2.58) and the proof of Theorem 2.4.

Theorem 2.5. The following result holds true:

Proof of Eq. (2.64): The use of (2.26), (2.27), (2.28) and (2.45) yields the following expression of $\mathcal{S}_{4}$

or in the terms of the hypergeometric function ${_2}F_{1}$ as

With the help of the Euler's transformation (see (2.32)), (2.68) can be rewritten as

This last summation in the above equation is the Humbert function $\Phi_{1}\left(a, n; c+n; \frac{x}{x-1}; \frac{xz}{1-x}\right)$. If one introduces this function, the result (2.64) is proved.

Proof of Eq. (2.65): If we apply the Euler's transformation on the function ${_2}F_{1}$ which appears in (2.69), we find

The last summation in (2.70) is the first Appell function $F_{1}\left(a, -, n, c+n; \frac{x}{x-1}, \frac{xz}{1-x}\right)$ (see (1.14)), which proves the desired result (2.65).

Proof of Eq. (2.66):

The series in (2.22) can be written as

By using the identity given by (2.48), (2.71) can be written as

The last summation in (2.72) is the third Appell function $F_{3}\left(a, a, c+s, b; c; x, y\right)$. This completes the proof of (2.66) and consequently of Theorem 2.5.

3.   Applications

In this section, we will apply our main results to express the output field resulting from the propagation of generalized Humbert Gaussian beams through a paraxial ABCD optical system. These beams are obtained by modulating generalized Humbert beams ^[3] with a Gaussian envelope.

The output field is given by Huygens-Fresnel integral ^[5]

where $k = 2\pi/\lambda$ is the wave number, $\lambda$ is the optical wavelength and $E(\rho^\prime, \phi^\prime)$ is the input field of the form

where $x = \frac{2i\alpha^{\prime}}{\alpha+i\alpha^{\prime}}$, $y = -\frac{k^2\rho^{\prime}2}{4z_0^2\left(\alpha+i\alpha^{\prime}\right) }$, $z_{0}$ is a constant, $a = l/2+m+1$, $b = \beta$, $c = m+l+1$ and $c^{\prime} = m+1$.

In (3.2), $C_1$ is a constant, $w_{0}$ is the waist width of the Gaussian part, $m$ is the beam order and $l$ is the topological charge.

By using the development of Humbert confluent Hypergeometric function in terms of Hypergeometric functions ${_2}F_{1}\left(a, b; c; x \right)$ ^[14]

and by substituting (3.2) in (3.1) with the help of the identity given by (2.9) and the well-known relationship ^[6]

The output field can be expressed in terms of Hypergeometric functions ${_1}F_{1}\left(a; b; x \right)$ and ${_2}F_{1}\left(a, b; c; x \right)\ $ as

where $C_Z$ is a function of the coordinate Z and $z = \frac{k^{2}\rho^{2}}{4\gamma B^2}$.

By using (2.19) of Theorem 2.2, (3.4) can be written in terms of the Humbert function $\Phi_{1}\left(a, b; c; x, y \right)$ as

or in terms of Appell function $F_{2}\left(a, b, b^{\prime}; c, c^{\prime}; x, y \right)$ as

In the following, we investigate some numerical simulations of particular cases of GHGBs called as $GHGBs_{sLGBs}$, $GHGBs_{QBGBs}$ and $GHGBs_{WGBs}$ which are derived from standard Laguerre-Gaussian beams, quadratic Bessel-Gaussian beams and Whittaker-Gaussian beams respectively after passing through a paraxial ABCD optical system with a spiral phase plate.

Figure 1 illustrates the normalized intensity distribution as a function of the transverse coordinate for particular cases of GHGBs propagating in free space system with two values of the lowest orders beam (m = 0 and m = 1), where $A = 1$, $B = z$, $C = 0$ and $D = 1$. From the plots of this figure, it’s seen that the GHGBs family is characterized by a dark spot at the center with zero intensity. It’s observed that when the beam orders increase, the side lobes of $GHGBs_{sLGBs}$ and $GHGBs_{WGBs}$ disappear.

4.   Conclusion

In this note, we have obtained a new general formulae for the integral transforms containing the product of Humbert and Bessel functions. For each confluent Appell function, we have evaluated the corresponding integral transfoms which is an infinite series of generalized hypergeometric and Appell functions. An application of our main results is investigated to evaluate the output generalized Humbert-Gaussian beam propagating through a paraxial ABCD optical system.

Conflict of interest

The authors declare that there is no conflict of interest.