Part 1. Infinite series and logarithmic integrals associated to differentiation with respect to parameters of the Whittaker $\mathrm{M}_{\kappa ,\mu }\left( x\right) $ function

First derivatives of the Whittaker function $\mathrm{M}_{\kappa ,\mu }\left(x\right) $ with respect to the parameters are calculated. Using the confluent hypergeometric function, these derivarives can be expressed as infinite sums of quotients of the digamma and gamma functions. Also, it is possible to obtain these parameter derivatives in terms of finite and infinite integrals with integrands containing elementary functions (products of algebraic, exponential and logarithmic functions) from the integral representation of $\mathrm{M}_{\kappa ,\mu }\left( x\right) $. These infinite sums and integrals can be expressed in closed-form for particular values of the parameters. For this purpose, we have obtained the parameter derivative of the incomplete gamma function in closed-form. As an application, reduction formulas for parameter derivatives of the confluent hypergeometric function has been derived, as well as some finite and infinite integrals containing products of algebraic, exponential, logarithmic and Bessel functions. Finally, some reduction formulas for the Whittaker functions $\mathrm{M}_{\kappa ,\mu }\left( x\right) $ and integral Whittaker functions $\mathrm{Mi}_{\kappa ,\mu }\left( x\right) $ and $\mathrm{mi}_{\kappa ,\mu }\left( x\right) $ are calculated.


Introduction
Introduced in 1903 by Whittaker [30], the M κ,µ (x) and W κ,µ (x) functions are defined as: where Γ (x) denotes the gamma function. These functions, called Whittaker functions, are closely associated to the following confluent hypergeometric function (Kummer function): where p F q a 1 , . . . , a p b 1 , . . . , b q x denotes the generalized hypergeometric function.
For particular values of the parameters κ and µ, the Whittaker functions can be reduced to a variety of elementary and special functions. Whittaker [30] discussed the connection of the functions defined in (1) and (2) with many other special functions, such as the modified Bessel function, the incomplete gamma functions, the parabolic cylinder function, the error functions, the logarithmic and the cosine integrals, and the generalize Hermite and Laguerre polynomials. Monographs and treatises dealing with special functions [10,13,16,[22][23][24]26,28,31] present the properties of the Whittaker functions with more or less extension.
The Whittaker functions are frequently applied in various areas of mathematical physics (see, for example [12,25,27]), such as the well-known solution of the Schrödinger equation for the harmonic oscillator [18]. M κ,µ (x) and W κ,µ (x) are usually treated as functions of variable x with fixed values of the parameters κ and µ. However, there are few investigations which consider κ and µ as variables. For instance, Laurenzi [19] discussed methods to calculate derivatives of M κ,1/2 (x) and W κ,1/2 (x) with respect to κ when this parameter is an integer. Using the Mellin transform, Buschman [11] showed that the derivatives of certain Whittaker functions with respect to the parameters can be expressed in finite sums of Whittaker functions. López and Sesma [21] considered the behaviour of M κ,µ (x) as a function of κ. They derived a convergent expansion in ascending powers of κ, and an asymptotic expansion in descending powers of κ. Using series of Bessel functions and Buchholz polynomials, Abad and Sesma [1] presented an algorithm for the calculation of the n-th derivative of the Whittaker functions with respect to parameter κ. Becker [6] investigated certain integrals with respect to parameter µ. Ancarini and Gasaneo [2] presented a general case of differentiation of generalized hypergeometric functions with respect to the parameters in terms of infinite series containing the digamma function. In addition, Sofostasios and Brychkov [29] considered derivatives of hypergeometric functions and classical polynomials with respect to the parameters.
In this paper, our main focus will be directed to the systematic investigation of the first derivatives of M κ,µ (x) with respect to the parameters. We will mainly base our results on two different approaches. The first one has to do with the series representation of M κ,µ (x), and the second one has to do with the integral representations of M κ,µ (x). Regarding the first approach, direct differentiation of (1) with respect to the parameters leads to infinite sums of quotients of digamma and gamma functions. It is possible to calculate such sums in closedform for particular values of the parameters. The parameter differentiation of the integral representations of M κ,µ (x) leads to finite and infinite integrals of elementary functions, such as products of algebraic functions, exponential and logarithmic functions. These integrals are similar to those investigated by Kölbig [17] and Geddes et al. [14]. As in the case of the first approach, it is possible to calculate such integrals in closed-form for some particular values of the parameters.
In the Appendices, we calculate the first derivative of the incomplete gamma functions γ (ν, x) and Γ (ν, x) with respect to the parameter ν. These results will be used when we calculate some of the integrals found in the second approach mentioned before. Also, we calculate some new reduction formulas of the integral Whittaker functions, which were recently introduced by us in [4]. They are defined in a similar way as other integral functions in the mathematical literature: Finally, we also include a list of reduction formulas for the Whittaker function M κ,µ (x) in the Appendices.
2 Parameter differentiation of M κ,µ via Kummer function 1 F 1 As mentioned before, the Whittaker function M κ,µ (x) is closely related to the confluent hypergeometric function 1 F 1 (a; b; x). Likewise, the parameter derivatives of M κ,µ (x) are also related to the parameter derivatives of 1 F 1 (a; b; x). Let us introduce the following notation set by Ancarini and Gasaneo [2].
Definition 1 Define the parameter derivatives of the confluent hypergeometric function as, and According to (3), we have x n n! , Since one of the integral representations of the confluent hypergeometric function is [22, Sect. 6.5.1]: Re b > Re a > 0, by direct differentiation of (8) with respect to parameters a and b, we obtain and Since the main focus is the systematic investigation of the parameter derivatives of M κ,µ (x), we will present these parameter derivatives as Theorems along the paper, and the corresponding results for G (1) (a; b; x) and H (1) (a; b; x) as Corollaries. Also, note that all the results regarding G (1) (a; b; x) can be transformed according to the next Theorem.

Theorem 2
The following transformation holds true: Proof. Differentiate with respect to a Kummer's transformation formula [24,Eqn. 13.2.39]: to obtain the desired result.
Corollary 4 For a ∈ R, a = 0, and for x ∈ R, the following reduction formula holds true: Proof. Direct differentiation of (1) yields thus comparing (14) with κ = −µ − 1 2 to (11), we arrive at (13), as we wanted to prove. Table 1 presents some explicit expressions for particular values of (11), and for x ∈ R, obtained with the help of MATHEMATICA program.
Next, we present other reduction formula of ∂M κ,µ (x) /∂κ from the result found in [19] for x ∈ R, where Ei (x) denotes the exponential integral and for n, ℓ = 1, 2, . . . and In order to calculate the finite sum given in (17), we derive the following Lemma.
Proof. Split the sum in two as Take a = 1, b = n, and z = 2 in the quadratic transformation [24,Eqn. 15.18 Now, apply Gauss's summation theorem [24,Eqn. 15.4.20] to arrive at Therefore, S 2 (n, ℓ) is a pure imaginary number. Since S (n, ℓ) is a real number, we conclude that S (n, ℓ) = Re [S 1 (n, ℓ)], as we wanted to prove.

Corollary 7
The following reduction formula holds true for n = 1, 2, . . . and x ∈ R, Proof. Consider (14) and (19) to arrive at the desired result.
In Table 2 we collect some particular cases of (19) for x ∈ R obtained with the help of MATHEMATICA program. Table 2: Derivative of M κ,µ with respect to κ by using (19).
Using (26), the derivative of M κ,µ (x) with respect µ has been calculated for particular values of κ and µ, with x ∈ R, using the help of MATHEMATICA, and they are presented in Table 3.
Note that for µ = −1/2, we obtain an indeterminate expression in (26). For this case, we present the following result.

Theorem 10
The following parameter derivative formula of M κ,µ (x) holds true for x ∈ R: where I ν (x) denotes the modified Bessel function. Table 3: Derivative of M κ,µ with respect to µ by using (26).
to obtain (30), as we wanted to prove. The order derivative of the modified Bessel function I µ (x) is given in terms of the Meijer-G function and the generalized hypergeometric function ∀ Re x > 0, µ ≥ 0 [15]: where K ν (x) is the modified Bessel function of the second kind ; or in terms of generalized hypergeometric functions only ∀ Re x > 0, µ > 0, µ / ∈ Z [7]: There are different expressions for the order derivatives of the Bessel functions [5,8]. This subject is summarized in [3], where general results are presented in terms of convolution integrals, and order derivatives of Bessel functions are found for particular values of the order.
Using (30), (32) and (33), some derivatives of M κ,µ (x) with respect µ has been calculated for x ∈ R with the help of MATHEMATICA, and they are presented in Table 4. Table 4: Derivative of M κ,µ with respect to µ by using (30).
denotes the beta function. In order to calculate the first derivative of M κ,µ (x) with respect to parameter κ, let us introduce the following finite logarithmic integrals.

Theorem 13
The following integral holds true for x ∈ R: Proof. Comparing (39) to (14), taking into account (1), we arrive at (49), as we wanted to prove.

Corollary 14
For κ = 0, Eqn. (49) is reduced to Theorem 15 For ℓ ∈ Z and m = 0, 1, 2, . . ., with m ≥ ℓ, the following integral holds true for x ∈ R: where F (s, k, z) Proof. From the definition of I 1 (κ, µ; x) given in (37), we have Perform the change of variables τ = 1 − t in the first integral above to arrive at where we have set Taking into account the binomial theorem and the integral (109) calculated in the Appendix, i.e.
1 0 e xt t m ln t dt = −1 to obtain According to [26, Eqn. 7.12.2(67)], we have that In order to obtain similar expressions as the ones obtained in Table 1, we derive an alternative form of (57). Indeed, from the definition of the hyperbolic sine and cosine integrals [24, Eqns. 6.2. [15][16], ∀z ∈ C, it is easy to prove that Chi (−z) = Chi (z) − ln z + ln (−z) .
it is easy to prove that thus, taking into account (58) and (59), we have Insert (60) in (57), to obtain Finally, substitute (61) in (56), and take into account (52), to arrive at Similarly, calculate Finally, according to (53), we arrive at (51), as we wanted to prove. Table 5 shows the integral I 1 (κ, µ; x) for x ∈ R and particular values of the parameters κ and/or µ, obtained from (49) and (51) with the aid of MATHE-MATICA program.

Corollary 18
For ℓ ∈ Z and m = 0, 1, 2, . . ., with m ≥ ℓ, the following reduction formula holds true for x ∈ R: Proof. Set (14) for κ = ℓ 2 and µ = m + 1−ℓ 2 and compare the result to (67). Table 6 shows the first derivative of M κ,µ (x) with respect to parameter κ for some particular values of κ and µ, and x ∈ R, which has been calculated from (67) and are not contained in Table 1.

Application to the calculation of infinite integrals
Additional integral representations of the Whittaker function M κ,µ (x) in terms of Bessel functions [22, Sect. 6.5.1] are known: Let us introduce the following infinite logarithmic integrals.

Derivative with respect to the second parameter ∂M κ,µ (x) /∂µ
In order to calculate the first derivative of M κ,µ (x) with respect to parameter µ, let us introduce the following finite logarithmic integrals.
Theorem 23 For ℓ ∈ Z and m = 0, 1, 2, . . ., with m ≥ ℓ, the following integral holds true for x ∈ R: Proof. From the definition of J 1 (κ, µ; x) given in (79), we have Perform the change of variables τ = 1 − t in the second integral above to arrive at where we follow the notation given in (54) for the integral I 1 (κ, µ; x). According to the results obtained in (55) and (62), we arrive at (88), as we wanted to prove.
Theorem 24 For ℓ ∈ Z and m = 0, 1, 2, . . ., with m ≥ ℓ, the following reduction formula holds true for x ∈ R: Proof. Insert (63) and (88) in (83) and apply (23). Table 7 shows the first derivative of M κ,µ (x) with respect to parameter µ for some particular values of κ and µ, andfor x ∈ R, which has been calculated from (90) and are not contained in Tables 3 and 4.

Application to the calculation of finite integrals
Theorem 26 For µ ≥ 0 and x ∈ R, the following finite integral holds true: where ∂I µ (x) /∂µ is given by (32) or (33).
Proof. Consider x > 0. Take κ = 0 in (85) and susbtitute (31) to obtain Now, insert in (98) the result given in (95) and simplify to get for x > 0 Finally, note that performing in (81) the change of variables τ = −t, we obtain the reflection formula so that from (99) and (100) we arrive at (97), as we wanted to prove. Table 8 shows the integral J 1 (κ, µ; x) for particular values of the parameters κ and µ, and x ∈ R, obtained from (87), (88) and (92) with the aid of MATHEMATICA program.

Conclusions
The Whittaker function M κ,µ (x) is defined in terms of the Kummer confluent hypergeometric function, hence its derivative with respect to the parameters κ and µ can be expressed as infinite sums of quotients of the digamma and gamma functions. Also, the parameter differentiation of the integral representations of M κ,µ (x) leads to finite and infinite integrals of elementary functions. These sums and integrals has been calculated for particular values of the parameters κ and µ in closed-form. As an application of these results, we have obtained some reduction formulas for the derivatives of the confluent Kummer function with respect to the parameters, i.e. G (1) (a, b; x) and H (1) (a, b; x). Also, we have calculated some finite integrals containing a combination of the exponential, logarithmic and algebraic functions, as well as some infinite integrals involving the exponential, logarithmic, algebraic and Bessel functions. It is worth noting that all the results presented in this paper has been both numerically and symbolically checked with MATHEMATICA program.
In the first Appendix, we have obtained the first derivative of the incomplete gamma functions in closed-form. These results allow us to calculate a finite logarithmic integral, which has been used to calculate one of the integrals appearing in the body of the paper.
In the second Appendix, we have calculated some new reduction formulas for the integral Whittaker functions Mi κ,µ (x) and mi κ,µ (x) from two reduction formulas of the Whittaker function M κ,µ (x). One of the latter seems not to be reported in the literature.
In the third Appendix we collect some reduction formulas for the Whittaker function M κ,µ (x).

A Parameter differentiation of the incomplete gamma functions
Definition 28 The lower incomplete gamma function is defined as [23]: The relation between both functions is The lower incomplete gamma function has the following series expansion [23, Eqns. 45:6:1] Also, the following integral representations in terms of infinite integrals hold true [24,Eqns. 8.6.3&7] for Re z > 0, From (101), the derivative of the lower incomplete gamma function with respect to the order ν has the following integral representation: Theorem 30 The parameter derivative of the lower incomplete gamma function is Proof. According to (101) and (104), the derivative of the lower incomplete gamma function with respect to the parameter ν is, Now, apply the sum formula [9, Eqn. 6.2.1(63)] to arrive at (106), as we wanted to prove.

Theorem 31 The parameter derivative of the upper incomplete gamma function is
Proof. Differentiate (103) with respect to the parameter ν and apply the result given in (106).
Corollary 32 From (105) and (106), we calculate the following integral: Corollary 33 The following integral holds true for x ∈ R: Proof. Perform the change of variables t = z τ in the integral given in (108), split the result in two integrals and apply again the change of variables t = x τ to the first integral, Comparing (108) to (110), we obtain (109), as we wanted to prove.
B Reduction formulas for integral Whittaker functions Mi κ,µ and mi κ,µ In [4], we found some reduction formulas for the integral Whittaker function Mi κ,µ (x). Next, we derive some new reduction formulas for Mi κ,µ (x) and mi κ,µ (x) from reduction formulas of the Whittaker function M κ,µ (x).
Proof. Follow similar steps as in the previous theorem, but consider the definition of the upper incomplete gamma function (102).
Theorem 38 The following reduction formula holds true for x ∈ R, n = 0, 1, 2, . . . and κ > 0: Apply (117)  Finally, take into account the defintion of the lower incomplete gamma function (101) and simplify the result to arrive at (116), as we wanted to prove.
Remark 39 It is worth noting that we could not locate the reduction formula (118) in the literature. C Reduction formulas for the Whittaker function M κ,µ (x) For convenience of the readers, reduction formulas for the Whittaker function M κ,µ (x) are presented in their explicit form in Table 9 for x ∈ R.