Part 2. Infinite series and logarithmic integrals associated to differentiation with respect to parameters of the Whittaker $\mathrm{W}_{\kappa ,\mu }\left( x\right) $ function

First derivatives with respect to the parameters of the Whittaker function $\mathrm{W}_{\kappa ,\mu }\left( x\right) $ 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 infinite integrals with integrands containing elementary functions (products of algebraic, exponential and logarithmic functions) from the integral representation of $\mathrm{W}_{\kappa ,\mu }\left( x\right) $. These infinite sums and integrals can be expressed in closed-form for particular values of the parameters. Finally, an integral representation of the integral Whittaker function $\mathrm{wi}_{\kappa ,\mu }\left( x\right) $ and its derivative with respect to $\kappa $, as well as some reduction formulas for the integral Whittaker functions $\mathrm{Wi}_{\kappa ,\mu }\left( x\right) $ and $\mathrm{wi}_{\kappa ,\mu }\left( x\right) $ are calculated.

Mostly, the Whittaker functions are regarded as a function of variable x with fixed values of parameters κ and µ, although there are few investigations where mathematical operations associated with both parameters are considered, especially for the κ parameter [1,6,10,17]. In this context, it is worthwhile to mention Laurenzi's paper [17], where the calculation of the derivative of W κ,1/2 (x) with respect to κ when this parameter is an integrer is derived. In [10], Buschman showed that the derivative of W κ,µ (x) with respect to the parameters can be expressed in terms of finite sums of these W κ,µ (x) functions. Higher derivatives of the Whittaker functions with respect to parameter κ were discussed by Abad and Sesma [1], and integrals with respect to parameter µ by Becker [6]. Since the Whittaker functions are related to the confluent hypergeometric function, it is worth mention the investigation of the derivatives of the generalized hypergeometric functions presented by Ancarini and Gasaneo [2] or Sofostasios and Brychkov [26].
The integral Whittaker functions were introduced by us [4] as follows: In the current paper, the main attention will be devoted to Whittaker function W κ,µ (x) by analyzing the first derivative of this function with respect to the parameters from the corresponding series and integral representations. Direct differentiation of the Whittaker functions leads to infinite sums of quotients of the digamma and gamma functions. It is possible to calculate these sums in closed-form in some cases with the aid of MATHEMATICA program. When the integral representations of the Whittaker function W κ,µ (x) are taken into account, the results of differentiation can be expressed in terms of Laplace transforms of elementary functions. Integrands of the these Laplace type integrals include products of algebraic, exponential and logarithmic functions. New groups of infinite integrals are comparable to those investigated by Kölbig [15], Geddes et al. [12], and Apelblat and Kravitzky [5] are calculated in this paper.
Also, we will focus our attention on the integral Whittaker functions Wi κ,µ (x) and wi κ,µ (x) in order to derive some new reduction formulas, as well as an integral representation of wi κ,µ (x) and its first derivative with respect to parameter κ.
2 Parameter differentiation of W κ,µ via Kummer function 1 F 1 Notation 1 Unless indicated otherwise, it is assumed throughout the paper that x is a real variable and z is a complex variable.
Next, we present other reduction formula of ∂W κ,µ (x) /∂κ from the result found in [17].
n (x) denotes the Laguerre polynomial.
In Table 2 we collect some particular cases of (22), obtained with the help of MATHEMATICA program.
Note that for n = 0, we obtain an indeterminate expression in (22). We calculate this particular case with a result of the next Section.

Theorem 5
The following reduction formula holds true: Proof. According to [22,Eqn. 13.18.2], we have thus, performing the derivative with respect to κ, Taking κ = 0 and considering (23), we have Finally, apply (31) and (33), to arrive at (27) as we wanted to prove.

2.2
Derivative with respect to the second parameter ∂W κ,µ (x) /∂µ Theorem 6 For 2µ / ∈ Z, the following parameter derivative formula of W κ,µ (x) holds true: Insert (13) in (30) to arrive at (29), as we wanted to prove. Table 3 shows the derivative of W κ,µ (x) with respect µ for particular values of κ and µ using (29) and the help of MATHEMATICA program.

Theorem 7
The following parameter derivative formula of W κ,µ (x) holds true: where K ν (x) denotes the modified Bessel of the second kind (Macdonald function).
Proof. Differentiate with respect to µ the expression [22, Eqn. 13.18.9]: to obtain as we wanted to prove. The order derivative of K µ (x) is given in terms of Meijer-G functions for Re x > 0, and µ ≥ 0 [13]: where I ν (x) is the modified Bessel function; or in terms of generalized hyper- Table 3: Derivative of W κ,µ with respect to µ by using (29).
geometric functions for Re x > 0, µ > 0, and 2µ / ∈ 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 (31), (33) and (34), some derivatives of W κ,µ (x) with respect µ has been calculated with the help of MATHEMATICA program, and they are presented in Table 4.
and as the infinite integral: In order to calculate the first derivative of W κ,µ (x) with respect to parameter κ, let us introduce the following finite logarithmic integrals.
Proof. Compare (10) to (39) and take into account (1) to arrive at (42), as we wanted to prove. Now, we derive a Lemma that will be applied throughout this Section and the next one.
Proof. Split the integral in two terms as follows: , and apply the Laplace transform for x > 0 [24, Eqn. 2.5.2(4)] 1 : to obtain Note that, according to Kummer's transformation (11), and to the reduction formula [24, Eqn. 7.11.1(14)]: we have for x > 0 It is worth noting that there is an incorrect sign in the reference cited. (45) and (47) in (78) to arrive at

Theorem 11
The following integral holds true for µ > 0 and x > 0: Proof. From (37) and (43), we obtain the desired result. (53), we obtain the following alternative form:

Remark 12 If we insert (48) in
x .
and the result given in (52) to arrive at (55).

Remark 14
If we consider (54), we obtain the following alternative form: x . Table 5 shows the first derivative of W κ,µ (x) with respect to parameter κ for some particular values of κ and µ, and x > 0, calculated with the aid of MATHEMATICA program from (57).
In order to prove (60), consider [ and performing the substitution where H n = n k=1 1 k denotes the n-th harmonic number. In order to prove (61), note that cot x = cot (x + π) and for x ∈ (−π, π) we have the expansion [21, Eqn. 44:6:2] Insert (64) in (59) to arrive at On the other hand, consider the reduction formula (128), derived in the Appendix, where E 1 (z) denotes the exponential integral [22, Eqn. 6.2.1], which is defined as where the path does not cross the negative real axis or pass throught the origin. Also, consider the property [22, Eqn. 6.2.4] Therefore, substituting (66) and (67) in (65), and taking into account (68), we arrive at (58), as we wanted to prove.

Remark 16
It is worth noting that from [10], where −N ≤ M ≤ N and M, N are integers of like parity, we can derive an equivalent reduction formula to (58). Indeed, taking N = M = m, (69) is reduced to (70) Note that from (28), we have Also, from (5) and the reduction formula for n = 0, 1, ... given in [22,Eqn. 13.2.8] U (a, a + n + 1, z) = z −a n s=0 n s (a) s z −s , Therefore, susbtituting (71) and (72) in (70), and simplifying, we arrive at Perform the index substitution s → s + k and exchange the sum order in (73), to arrive at By virtue of the binomial theorem, the inner sum in (74) is just −1, thus we finally obtain: Theorem 17 For n = 0, 1, 2, . . ., and x > 0, the following integral holds true: Proof. From (37), we have thus, taking µ = n+1 2 with n = 0, 1, 2, . . . and applying the binomial theorem, we get Insert the result obtained in (43) for ν = k in (77) to arrive at Now, take into account (66), to get Finally, note that using the exponential polynomial, defined as e n (x) = n k=0 x k k! , and the property for n = 0, 1, 2, . . . [21,Eqn. 45:4:2]: we calculate the following finite sum as: Apply (79) to (78) in order to obtain (76), as we wanted to prove.

Application to the calculation of infinite integrals
Additional integral representations of the Whittaker function W κ,µ (x) in terms of Bessel functions [19,Sect. 7.4.2] are known: Let us introduce the following infinite logarithmic integral.

Remark 27 If we take into account (48) in
thus for −2µ = 0, 1, 2, . . . and x > 0, we have x + ψ (2µ) − ln x . Table 7 shows the first derivative of W κ,µ (x) with respect to parameter µ for some particular values of κ and µ, with x > 0, calculated from (103) with the aid of MATHEMATICA program.
Remark 37 It is worth noting that we cannot follow the above steps to derive the integral representation of Wi κ,µ (x) because the corresponding integral does not converge, except for some special cases such as the ones given in (114).

Conclusions
The Whittaker function W κ,µ (x) is defined in terms of the Tricomi 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 some integral representations of W κ,µ (x) leads to infinite integrals of elementary functions. These sums and integrals has been calculated for some particular cases of the parameters κ and µ in closed-form.
As an application of these results, we have calculated an infinite integral containing the Macdonald function. It is worth noting that all the results presented in this paper has been both numerically and symbolically checked with MATH-EMATICA program.
In the first Appendix, we calculate a reduction formula for the first derivative of the Kummer function, i.e. G (1) (a; a; z), which it is necessary for the derivation of Theorem 3.
In the second Appendix, we calculate a reduction formula of the hypergeometric function 2 F 2 (1, 1; 2, 2 + m; x) for non-negative integer m, since it is not found in most common literature, such as [24]. This reduction formula is used throughout Section 3 in order to simplify the results obtained.
Finally, we collect some reduction formulas for the Whittaker function W κ,µ (x) in the last Appendix.
A Calculation of G (1) (a; a; z) Theorem 39 The following reduction formula holds true: a a x = x e x a 2 F 2 1, 1 a + 1, 2 − x .
B Calculation of 2 F 2 (1, 1; 2, 2 + m; x) Theorem 40 For m = 0, 1, 2, . . ., the following reduction formula holds true:  to arrive at (128), as we wanted to prove C Reduction formulas for the Whittaker function W κ,µ (x) For convenience of the readers, reduction formulas for the Whittaker function W κ,µ (x) are presented in their explicit form in Table 9.