Euler Type Integrals and Integrals in Terms of Extended Beta Function

Abstract

We derive the evaluations of certain integrals of Euler type involving generalized hypergeometric series. Further, we establish a theorem on extended beta function, which provides evaluation of certain integrals in terms of extended beta function and certain special polynomials. The possibility of extending some of the derived results to multivariable case is also investigated.

1. Introduction

Euler generalized the factorial function from the domain of natural numbers to the gamma function Γ(𝛼)=^∞₀𝑡^𝛼−1𝑒^−𝑡𝑑𝑡(Re(𝛼)>0), (1) defined over the right half of the complex plane. This led Legendre (in 1811) to decompose the gamma function into the incomplete gamma functions, 𝛾(𝛼,𝑥) and Γ(𝛼,𝑥) [1],𝛾(𝛼,𝑥)=^𝑥₀𝑡^𝛼−1𝑒^−𝑡𝑑𝑡(Re(𝛼)>0),Γ(𝛼,𝑥)=^∞_𝑥𝑡^𝛼−1𝑒^−𝑡𝑑𝑡, (2a)(2b)which are obtained from (1) by replacing the upper and lower limits by 𝑥, respectively. The closed-form solutions to a considerable number of problems in applied mathematics, astrophysics, nuclear physics, statistics, and engineering can be expressed in terms of incomplete gamma functions. These functions develop singularities at the negative integers. Chaudhry and Zubair [2] extended the domain of these functions to the entire complex plane by inserting a regularization factor exp(−𝑏/𝑡) in the integrand of (1). For Re(𝑏)>0, this factor clearly removes the singularity coming from the limit 𝑡=0. For 𝑏=0, this factor becomes unity, and thus we get the original gamma function. We note the following relation [3, page 20 (1.2)]: Γ_𝑏(𝛼)=^∞₀𝑡^𝛼−1𝑏exp−𝑡−𝑡(𝑏)𝑑𝑡=2^𝛼/2𝐾_𝛼2√𝑏(Re(𝛼)>0), (3) where 𝐾_𝑛(𝑥) is the modified Bessel function of the second kind of order 𝑛 (or Macdonald’s function) [1]. The relationships between the generalized gamma and Macdonald functions could not have been apparent in the original gamma function. These generalized gamma functions proved very useful in diverse engineering and physical problems; see, for example, [2, 3] and references therein.

We note that Riemann’s zeta function 𝜁(𝑥) defined by the series [1, page 85 (2.98)] 𝜁(𝑥)=^∞_𝑛=11𝑛^𝑥(𝑥>1) (4) is useful in proving convergence or divergence of other series by means of comparison test. Zeta function is closely related to the logarithm of the gamma function and to the polygamma functions. The regularizer exp(−𝑏/𝑡) also proved very useful in extending the domain of Riemann’s zeta function, thereby providing relationships that could not have been obtained with the original zeta function.

In view of the fact that the regularization factor is useful in extending the domain of the gamma and zeta functions. The domain of other special functions could be usefully extended in a similar manner.

In particular, Euler’s beta function 𝐵(𝛼,𝛽) has a close relationship to gamma function, that is, 𝐵=Γ(𝛼)Γ𝛽𝛼,𝛽=𝐵𝛽,𝛼Γ,𝛼+𝛽 (5) and can be usefully extendable. We first recall the basic Euler integral which defines the beta function as follows: 𝐵𝛼,𝛽=¹₀𝑢^𝛼−1(1−𝑢)^𝛽−1𝑑𝑢Re(𝛼),Re𝛽>0. (6)

Keeping in view the fact that for the 𝛼-𝛽 symmetry to be preserved there must be symmetry of the integrand in 𝑢 and 1−𝑢, Chaudhry et al. [3] introduced an extension of Euler’s beta function 𝐵(𝛼,𝛽) in the following form: 𝐵𝛼,𝛽;𝑏=¹₀𝑡^𝛼−1(1−𝑡)^𝛽−1−𝑏exp𝑡()((𝑏)),1−𝑡𝑑𝑡Re>0 (7) which for 𝑏=0 reduces to the original beta function.

The transformations 𝑡=(𝑢+1)/2 and 𝑡=(𝑢−𝑎)/(𝑐−𝑎) in (7) yield the following integral representations for the extended beta function (EBF) 𝐵(𝛼,𝛽;𝑏) [3, page 22 (2.9), (2.11)]: 𝐵𝛼,𝛽;𝑏=2^{1−𝛼−𝛽}×¹₋₁(1+𝑢)^𝛼−1(1−𝑢)^𝛽−1exp−4𝑏1−𝑢²((𝑏)),𝑑𝑢Re>0𝐵=()𝛼,𝛽;𝑏𝑐−𝑎^{1−𝛼−𝛽}×^𝑐_𝑎(𝑢−𝑎)^𝛼−1(𝑐−𝑢)^𝛽−1−𝑏()exp𝑐−𝑎²()((𝑏)),𝑢−𝑎)(𝑐−𝑢𝑑𝑢Re>0 (8)(9) respectively.

The EBF 𝐵(𝛼,𝛽;𝑏) is extremely useful in the sense that most of the properties of the beta function carry over naturally and simply for it. This extension is also important due to the fact that this function is related to other special functions for particular values of the variables.

We note the following connections [3]: 𝐵(𝛼,−𝛼;𝑏)=2𝑒^−2𝑏𝐾_𝛼(2𝑏)(Re(𝑏)>0),𝐵()=√𝛼,𝛼;𝑏𝜋2^−𝛼𝑏^(𝛼−1)/2𝑒^−2𝑏𝑊_{−𝛼/2,𝛼/2}()=√4𝑏𝜋2^1−2𝛼𝑒^−4𝑏Ψ12,1−𝛼;4𝑏(Re(𝑏)>0), (10)(11) where 𝑊_𝑚,𝑛(𝑧) denotes the Whittaker function [1] and Ψ(𝑎,𝑐;𝑧) denotes the second form of solutions of Kummer’s equation [4].

Also, we note that 𝐵(0,0;𝑏)=2𝑒^−2𝑏𝐾₀((𝑏)),𝐵12𝑏)(Re>021,−2;𝑏=𝜋𝑏𝑒^−4𝑏((𝑏)),𝐵1Re>02,12;𝑏=𝜋Erfc2√𝑏(Re(𝑏)>0), (12) where Erfc(𝑧)=(√𝜋/2)erfc(𝑧) denotes the complementary error function [4].

Explicit evaluations of some integrals of Euler type ¹₀𝑢^𝛼−1(1−𝑢)^𝛽−1𝑓(𝑢)𝑑𝑢 (13) for some particular functions 𝑓, specially in the symmetric case 𝛼=𝛽, are derived in [5]. These evaluations are related to various reduction formulae for hypergeometric functions represented by such integrals. These formulae generalize the evaluations of some symmetric Euler integrals implied by the following result due to Pitman [6].

If a standard Brownian bridge is sampled at time 0, time 1 and at 𝑛 independent random times with uniform distribution on [0,1], then the broken line approximation to the bridge obtained from these 𝑛+2 values has a total variation whose mean square is 𝑛(𝑛+1)/(2𝑛+1).

Motivated and inspired by the work of Ismail and Pitman [5] and Chaudhry et al. [3], in this paper, we derive the evaluations of certain Euler type integrals and some integrals in terms of EBF 𝐵(𝛼,𝛽;𝑏). In Section 2, we obtain the evaluations of certain integrals of the following type: 𝐼_{𝑑,𝛼,𝛽,𝛾,𝑎,𝑒}𝜙(𝑡)(𝑡=1,𝜓)𝐵𝛼,𝛽^𝑒_𝑎(𝑡−𝑎)^𝛼−1(𝑒−𝑡)^𝛽−1𝜙(𝑡)^𝛾×_𝑟𝐹_𝑡𝑓_𝑟;(𝑡)𝑔𝑑𝜓_𝑡;𝑑𝑡, (14) where _𝑟𝐹_𝑡 denotes the generalized hypergeometric series [4], for particular functions 𝜙(𝑡) and 𝜓(𝑡). In Section 3, we establish a theorem on EBF 𝐵(𝛼,𝛽;𝑏) and apply it to obtain evaluations of certain integrals in terms of EBF 𝐵(𝛼,𝛽;𝑏). Finally, we give some concluding remarks in Section 4.

2. Euler Type Integrals

We derive the evaluations of certain integrals of the following type: 𝐼_{𝑑,𝛼,𝛽,𝛾,𝑎,𝑒}𝜙(𝑡)(𝑡=1,𝜓)𝐵𝛼,𝛽^𝑒_𝑎(𝑡−𝑎)^𝛼−1(𝑒−𝑡)^𝛽−1𝜙(𝑡)^𝛾×_𝑟𝐹_𝑡𝑓_𝑟;(𝑡)𝑔𝑑𝜓_𝑡;𝑑𝑡. (15)

We evaluate the integrals of type (15) for particular functions 𝜙(𝑡) and 𝜓(𝑡) by considering the following cases.

Case 1. Taking 𝜙(𝑡)=(1−𝑥₁𝑡)^−𝛼1(1−𝑥₂𝑡)^−𝛼2, 𝜓(𝑡)=1/𝑡(1−𝑡), 𝑎=0, and 𝑒=𝛾=1 in (15) and using [5, page 962 (7)] ¹₀𝑡^𝛼−1(1−𝑡)^𝛽−11−𝑥₁𝑡^−𝛼11−𝑥₂𝑡^−𝛼2𝐹𝑑𝑡=𝐵𝛼,𝛽₁𝛼,𝛼₁,𝛼₂;𝛼+𝛽;𝑥₁,𝑥₂Re(𝛼),Re𝛽>0;max𝑥₁,𝑥₂<1, (16) where 𝐹₁ denotes the Appell function [4], in the r.h.s. after expanding _𝑟𝐹_𝑡^{(𝑓𝑟);𝑑𝜓(𝑡)}_(𝑔𝑡); and simplifying, we find the following integral:𝐼_{𝑑,𝛼,𝛽,1,0,1}1−𝑥₁𝑡^−𝛼11−𝑥₂𝑡^−𝛼2,1𝑡()=1−𝑡_{∞𝑚1,𝑚2=0}(𝛼)_𝑚1+𝑚2𝛼_1𝑚1𝛼_2𝑚2𝑥^𝑚1₁𝑥^𝑚2₂𝛼+𝛽_𝑚1+𝑚2𝑚₁!𝑚₂!×_𝑟+2𝐹_𝑡+2𝑓_𝑟,121−𝛼−𝛽−𝑚₁−𝑚₂1,1−2𝛼+𝛽+𝑚₁+𝑚₂;𝑔4𝑑_𝑡,1−𝛼+𝑚₁+𝑚₂(𝛼)𝛽𝑥,1−𝛽;Re,Re>0;max₁,𝑥₂<1. (17)

Case 2. Taking 𝜙(𝑡)=_𝑝𝐹_𝑞^{(𝑎𝑝);𝑐𝑡(𝑒−𝑡)}_(𝑏𝑞);, 𝜓(𝑡)=1/𝑡(𝑒−𝑡), 𝑎=0, and 𝛾=1 in (15) and using [7, page 303 (1)] ^𝑒₀𝑡^𝛼−1(𝑒−𝑡)_𝑝^𝛽−1𝐹_𝑞𝑎_𝑝;()𝑏𝑐𝑡𝑒−𝑡_𝑞;=(𝑒)𝑑𝑡_{𝛼+𝛽−1}𝐵×𝛼,𝛽_𝑝+2𝐹_𝑞+2𝑎_𝑝,𝛼,𝛽;𝑐𝑒²4𝑏_𝑞,𝛼+𝛽2,𝛼+𝛽+12;𝑒,Re(𝛼),Re𝛽>0;arg4−𝑐𝑒²<𝜋, (18) we find the following integral: 𝐼_{𝑑,𝛼,𝛽,1,0,𝑒𝑝}𝐹_𝑞𝑎_𝑝;()𝑏𝑐𝑡𝑒−𝑡_𝑞;,1𝑡()=(𝑒)𝑒−𝑡_{∞𝛼+𝛽−1𝑚=0}𝑎_𝑝𝑚(𝛼)_𝑚𝛽_𝑚𝑐𝑒²/4^𝑚𝑏_𝑞𝑚(𝛼+𝛽)/2_𝑚(𝛼+𝛽+1)/2_𝑚×𝑚!_𝑟+2𝐹_𝑡+2𝑓_𝑟,1211−𝛼−𝛽−𝑚,1−2𝛼+𝛽−𝑚;4𝑑𝑒²𝑔_𝑡(𝛼)𝛽,1−𝛼−𝑚,1−𝛽−𝑚;𝑒,Re,Re>0;arg4−𝑐𝑒²<𝜋. (19)

Case 3. Taking 𝜙(𝑡)=_𝑝𝐹_𝑞^{(𝑎𝑝);𝑐𝑡(𝑒−𝑡)}_(𝑏𝑞);, 𝜓(𝑡)=1/(𝑡²(𝑒−𝑡)²), 𝑎=0, and 𝛾=1 and proceeding as in Case 2, we find the following integral:𝐼_{𝑑,𝛼,𝛽,1,0,𝑒𝑝}𝐹_𝑞𝑎_𝑝;()𝑏𝑐𝑡𝑒−𝑡_𝑞;,1𝑡²(𝑒−𝑡)²=(𝑒)^{∞𝛼+𝛽−1}_𝑚=0𝑎_𝑝𝑚(𝛼)_𝑚𝛽_𝑚𝑐𝑒²/4^𝑚𝑏_𝑞𝑚𝛼+𝛽/2_𝑚𝛼+𝛽+1/2_𝑚×𝑚!_𝑟+6𝐹_𝑡+8𝑓_𝑟,𝑗−𝛼−𝛽2,1211−𝛼−𝛽−𝑚,1−2𝛼+𝛽−𝑚;4𝑑𝑒⁴𝑔_𝑡,𝑗−𝛼−𝛽4,1−𝛼−𝑚2,2−𝛼−𝑚2,1−𝛽−𝑚2,2−𝛽−𝑚2;𝑗=1,2,3,4𝑒,Re(𝛼),Re𝛽>0;arg4−𝑐𝑒²<𝜋. (20)

Case 4. Taking 𝜙(𝑡)=𝐊(𝑐𝑡(𝑒−𝑡)), 𝜓(𝑡)=1/𝑡(𝑒−𝑡), 𝑎=0, and 𝛾=1 in (15), where 𝐊(𝑧) is the complete elliptic integral of the first kind [4, page 35 (32)], defined by 𝐊(𝑧)=₀^𝜋/2𝑑𝜃1−𝑧²sin²𝜃 (21) and using [7, page 265 (4)] ^𝑒₀𝑡^𝛼−1(𝑒−𝑡)^𝛽−1𝐊𝑐𝑡()=𝜋𝑒−𝑡𝑑𝑡2(𝑒)^{𝛼+𝛽−1}𝐵𝛼,𝛽₄𝐹₃12,12𝑐,𝛼,𝛽;₂𝑒²41,𝛼+𝛽2,𝛼+𝛽+12;Re(𝛼),Re𝛽>0;arg4−𝑐²𝑒²<𝜋, (22) we find the following integral: 𝐼_{𝑑,𝛼,𝛽,1,0,𝑒}𝐊𝑐𝑡(),1𝑒−𝑡𝑡()=𝜋𝑒−𝑡2(𝑒)^{∞𝛼+𝛽−1}_𝑚=0(1/2)_𝑚(1/2)_𝑚(𝛼)_𝑚𝛽_𝑚𝑐²𝑒²/4^𝑚(1)_𝑚(𝛼+𝛽)/2_𝑚(𝛼+𝛽+1)/2_𝑚×𝑚!_𝑟+2𝐹_𝑡+2𝑓_𝑟,1211−𝛼−𝛽−𝑚,1−2𝛼+𝛽−𝑚;4𝑑𝑒²𝑔_𝑡(𝛼)𝛽,1−𝛼−𝑚,1−𝛽−𝑚;Re,Re>0;arg4−𝑐²𝑒²<𝜋. (23)

Case 5. Taking 𝜙(𝑡)=𝐄(𝑐𝑡(𝑒−𝑡)), 𝜓(𝑡)=1/𝑡(𝑒−𝑡), 𝑎=0, and 𝛾=1 in (15), where 𝐄(𝑧) is the complete elliptic integral of the second kind [4, page 35 (33)], defined by 𝐄(𝑧)=₀^𝜋/21−𝑧²sin²𝜃𝑑𝜃 (24) and using [7, page 280 (3)] ^𝑒₀𝑡^𝛼−1(𝑒−𝑡)^𝛽−1𝐄𝑐𝑡()=𝜋𝑒−𝑡𝑑𝑡2(𝑒)^{𝛼+𝛽−1}𝐵𝛼,𝛽₄𝐹₃−12,12𝑐,𝛼,𝛽;₂𝑒²41,𝛼+𝛽2,𝛼+𝛽+12;𝑒,Re(𝛼),Re𝛽>0;arg4−𝑐²𝑒²<𝜋, (25) we find the following integral: 𝐼_{𝑑,𝛼,𝛽,1,0,𝑒}𝐄𝑐𝑡(),1𝑒−𝑡𝑡()=𝜋𝑒−𝑡2(𝑒)^{∞𝛼+𝛽−1}_𝑚=0(−1/2)_𝑚(1/2)_𝑚(𝛼)_𝑚𝛽_𝑚𝑐²𝑒²/4^𝑚(1)_𝑚(𝛼+𝛽)/2_𝑚(𝛼+𝛽+1)/2_𝑚×𝑚!_𝑟+2𝐹_𝑡+2𝑓_𝑟,1211−𝛼−𝛽−𝑚,1−2𝛼+𝛽−𝑚;4𝑑𝑒²𝑔_𝑡(𝛼)𝛽,1−𝛼−𝑚,1−𝛽−𝑚;𝑒,Re,Re>0;arg4−𝑐²𝑒²<𝜋. (26)

Case 6. Taking 𝜙(𝑡)=arcsin(𝑐𝑡(𝑒−𝑡)), 𝜓(𝑡)=1/𝑡(𝑒−𝑡), 𝑎=0, and 𝛾=2 in (15) and using [7, page 173 (151)] ^𝑒₀𝑡^𝛼−1(𝑒−𝑡)^𝛽−1arcsin²𝑐𝑡(𝑒−𝑡)𝑑𝑡=𝑐²(𝑒)^𝛼+𝛽+1𝐵×𝛼+1,𝛽+1₅𝐹₄𝑐1,1,1,𝛼+1,𝛽+1;₂𝑒²432,2,𝛼+𝛽+22,𝛼+𝛽+32;𝑒>0;Re(𝛼),Re𝛽>−1;arg4−𝑐²𝑒²<𝜋, (27) we find the following integral: 𝐼_{𝑑,𝛼,𝛽,2,0,𝑒}arcsin𝑐𝑡(),1𝑒−𝑡𝑡()=𝑒−𝑡𝛼𝛽𝑐²(𝑒)^𝛼+𝛽+1×𝛼+𝛽𝛼+𝛽+1_∞𝑚=0(1)_𝑚(1)_𝑚(1)_𝑚(𝛼+1)_𝑚𝛽+1_𝑚𝑐²𝑒²/4^𝑚(3/2)_𝑚(2)_𝑚(𝛼+𝛽+2)/2_𝑚(𝛼+𝛽+3)/2_𝑚×𝑚!_𝑟+2𝐹_𝑡+2𝑓_𝑟1,−211+𝛼+𝛽−𝑚,−2𝛼+𝛽−𝑚;4𝑑𝑒²𝑔_𝑡(𝛼)𝛽,−𝛼−𝑚,−𝛽−𝑚;𝑒>0;Re,Re>−1;arg4−𝑐²𝑒²<𝜋. (28)

Case 7. Taking 𝜙(𝑡)=exp(𝑐²𝑡(𝑒−𝑡))erf(𝑐𝑡(𝑒−𝑡)), 𝜓(𝑡)=1/𝑡(𝑒−𝑡), 𝑎=0, and 𝛾=1 in (15) and using [7, page 185 (1)] ^𝑒₀𝑡^𝛼−1(𝑒−𝑡)^𝛽−1exp𝑐²𝑡(𝑒−𝑡)erf𝑐𝑡()=2𝑒−𝑡𝑑𝑡√𝜋𝑐(𝑒)^𝛼+𝛽𝐵1𝛼+21,𝛽+2×₃𝐹₃11,𝛼+21,𝛽+2;𝑐²𝑒²432,𝛼+𝛽+12,𝛼+𝛽+22;(𝛼)𝛽1𝑒>0;Re,Re>−2, (29) where erf(𝑧)=(2/√𝜋)Erf(𝑧) denotes the error function [4], we find the following integral: 𝐼_{𝑑,𝛼,𝛽,1,0,𝑒}exp𝑐²𝑡(𝑒−𝑡)erf𝑐𝑡(),1𝑒−𝑡𝑡()=2𝑒−𝑡√𝜋𝑐(𝑒)^𝛼+𝛽𝐵𝛼+(1/2),𝛽+(1/2)𝐵×𝛼,𝛽_∞𝑚=0(1)_𝑚(𝛼+(1/2))_𝑚𝛽+(1/2)_𝑚𝑐²𝑒²/4^𝑚(3/2)_𝑚(𝛼+𝛽+1)/2_𝑚(𝛼+𝛽+2)/2_𝑚×𝑚!_𝑟+2𝐹_𝑡+2𝑓_𝑟,1211−𝛼−𝛽−𝑚,−2𝛼+𝛽−𝑚;4𝑑𝑒²𝑔_𝑡,121−𝛼−𝑚,2(𝛼)𝛽1−𝛽−𝑚;𝑒>0;Re,Re>−2. (30) Now, we establish a theorem and apply it to obtain the evaluations of certain integrals in terms of extended beta function.

3. Integrals in Terms of Extended Beta Function

Consider an (𝑟+1)-variable generating function 𝐺(𝑥₁,𝑥₂,…,𝑥_𝑟;𝑡) which possesses a formal (not necessarily convergent for 𝑡≠0) power series expansion in 𝑡 such that [4, page 80 (8)] 𝐺𝑥₁,𝑥₂,…,𝑥_𝑟;𝑡=^∞_𝑛=0𝑐_𝑛𝑔_𝑛𝑥₁,𝑥₂,…,𝑥_𝑟𝑡^𝑛, (31) where each member of the generated set {𝑔_𝑛(𝑥₁,𝑥₂,…,𝑥_𝑟)}^∞_𝑛=0 is independent of 𝑡 and the coefficient set {𝑐_𝑛}^∞_𝑛=0 may contain the parameters of the set {𝑔_𝑛(𝑥₁,𝑥₂,…,𝑥_𝑟)}^∞_𝑛=0 but is independent of 𝑥₁,𝑥₂,…,𝑥_𝑟 and 𝑡.

Theorem 1. Let the generating function 𝐺(𝑥₁,𝑥₂,…,𝑥_𝑟;𝑡) defined by (31) be such that 𝐺(𝑥₁,𝑥₂,…,𝑥_𝑟;𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎) remains uniformly convergent for 𝑢∈(𝑎,𝑐), 𝜌,𝜎≥0, and 𝜌+𝜎>0. Then ^𝑐_𝑎(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^{𝜇−𝜆−1}×𝐺𝑥₁,𝑥₂,…,𝑥_𝑟;𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎−𝑏()×exp𝑐−𝑎²()=𝑢−𝑎)(𝑐−𝑢𝑑𝑢_∞𝑛=0(𝑐−𝑎)^{𝜇+(𝜌+𝜎)𝑛−1}𝑐_𝑛𝑔_𝑛𝑥₁,𝑥₂,…,𝑥_𝑟×𝐵𝜆+𝜌𝑛,𝜇−𝜆+𝜎𝑛;𝑏𝑡^𝑛Re𝜇>Re(𝜆)>0;Re(𝑏)>0, (32) where 𝐵(𝑥,𝑦;𝑏) is the extended beta function defined by (7).

Proof. Applying the definition of 𝐺(𝑥₁,𝑥₂,…,𝑥_𝑟;𝑡) given in (31) in the l.h.s. of (32), we get ^∞_𝑛=0𝑐_𝑛𝑔_𝑛𝑥₁,𝑥₂,…,𝑥_𝑟𝑡^𝑛×^𝑐_𝑎(𝑢−𝑎)^{𝜆+𝜌𝑛−1}(𝑐−𝑢)^{𝜇−𝜆+𝜎𝑛−1}−𝑏()×exp𝑐−𝑎²(𝑢−𝑎)(𝑐−𝑢)𝑑𝑢, (33) which by using (9) yields the r.h.s. of (32).

Corollary 2. With definition (31) and notations as in Theorem 1, one has ¹₀𝑢^𝜆−1(1−𝑢)^{𝜇−𝜆−1}𝐺𝑥₁,𝑥₂,…,𝑥_𝑟;𝑡𝑢^𝜌(1−𝑢)^𝜎−𝑏×exp𝑢()=1−𝑢𝑑𝑢_∞𝑛=0𝑐_𝑛𝑔_𝑛𝑥₁,𝑥₂,…,𝑥_𝑟𝐵𝜆+𝜌𝑛,𝜇−𝜆+𝜎𝑛;𝑏𝑡^𝑛Re𝜇>Re(𝜆)>0;Re(𝑏)>0. (34)

Proof. Taking 𝑎=0 and 𝑐=1 in (32), we get (34).

Corollary 3. With definition (31) and notations as in Theorem 1, one has ¹₋₁(1+𝑢)^𝜆−1(1−𝑢)^{𝜇−𝜆−1}×𝐺𝑥₁,𝑥₂,…,𝑥_𝑟;𝑡(1+𝑢)^𝜌(1−𝑢)^𝜎×exp−4𝑏1−𝑢²=𝑑𝑢_∞𝑛=0(2)^{𝜇+(𝜌+𝜎)𝑛−1}𝑐_𝑛𝑔_𝑛𝑥₁,𝑥₂,…,𝑥_𝑟×𝐵𝜆+𝜌𝑛,𝜇−𝜆+𝜎𝑛;𝑏𝑡^𝑛Re𝜇>Re(𝜆)>0;Re(𝑏)>0. (35)

Proof. Taking 𝑎=−1 and 𝑐=1 in (32), we get (35).

Corollary 4. With definition (31) and notations as in Theorem 1, one has ^𝑐_𝑎(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^−𝜆−1𝐺𝑥₁,𝑥₂,…,𝑥_𝑟;𝑡𝑐−𝑢𝑢−𝑎^𝜎−𝑏()×exp𝑐−𝑎²()=2𝑢−𝑎)(𝑐−𝑢𝑑𝑢(𝑐−𝑎)𝑒^∞2𝑏_𝑛=0𝑐_𝑛𝑔_𝑛𝑥₁,𝑥₂,…,𝑥_𝑟𝐾_{𝜆−𝜎𝑛}(2𝑏)𝑡^𝑛(Re(𝜆)>0;Re(𝑏)>0). (36)

Proof. Taking 𝜇=0 and 𝜌=−𝜎 in (32) and making use of relation (10), we get (36).
Now, we apply Theorem 1 to derive the evaluations of certain Euler type integrals in terms of EBF 𝐵(𝛼,𝛽;𝑏). We consider the following cases.
Case 1. Consider the generating function of the multivariable Hermite polynomials 𝐻_𝑛^{(1,2,…,𝑟)}(𝑥₁,𝑥₂,…,𝑥_𝑟) [8, page 602 (20,21)] exp𝑥₁𝑡+𝑥₂𝑡²+⋯+𝑥_𝑟𝑡^𝑟=^∞_𝑛=0𝐻_𝑛^{(1,2,…,𝑟)}𝑥₁,𝑥₂,…,𝑥_𝑟𝑡^𝑛.𝑛! (37)
Applying Theorem 1 to generating function (37), we find the following integral: ^𝑐_𝑎(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^{𝜇−𝜆−1}−𝑏()×exp𝑐−𝑎²()+𝑢−𝑎)(𝑐−𝑢_𝑟𝑗=1𝑥_𝑗𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎𝑗=𝑑𝑢_∞𝑛=0(𝑐−𝑎)^{𝜇+(𝜌+𝜎)𝑛−1}𝐻_𝑛^{(1,2,…,𝑟)}𝑥₁,𝑥₂,…,𝑥_𝑟𝑡×𝐵𝜆+𝜌𝑛,𝜇−𝜆+𝜎𝑛;𝑏^𝑛𝜇(𝜆)(𝑏).𝑛!Re>Re>0;Re>0;𝜌,𝜎≥0;𝜌+𝜎>0 (38)
Taking 𝑟=3 in (38), we get ^𝑐_𝑎(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^{𝜇−𝜆−1}−𝑏()×exp𝑐−𝑎²()+𝑢−𝑎)(𝑐−𝑢_3𝑗=1𝑥_𝑗𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎𝑗=𝑑𝑢_∞𝑛=0(𝑐−𝑎)^{𝜇+(𝜌+𝜎)𝑛−1}𝐻_𝑛𝑥₁,𝑥₂,𝑥₃𝑡×𝐵𝜆+𝜌𝑛,𝜇−𝜆+𝜎𝑛;𝑏^𝑛𝜇(𝜆)(𝑏),𝑛!Re>Re>0;Re>0;𝜌,𝜎≥0;𝜌+𝜎>0 (39) where 𝐻_𝑛(𝑥₁,𝑥₂,𝑥₃) are the 3-variable Hermite polynomials (3VHP) [9].
Again, taking 𝑟=2 in (38), we get ^𝑐_𝑎(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^{𝜇−𝜆−1}−𝑏()×exp𝑐−𝑎²()+𝑢−𝑎)(𝑐−𝑢_2𝑗=1𝑥_𝑗𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎𝑗=𝑑𝑢_∞𝑛=0(𝑐−𝑎)^{𝜇+(𝜌+𝜎)𝑛−1}𝐻_𝑛𝑥₁,𝑥₂𝑡×𝐵𝜆+𝜌𝑛,𝜇−𝜆+𝜎𝑛;𝑏^𝑛𝜇(𝜆)(𝑏),𝑛!Re>Re>0;Re>0;𝜌,𝜎≥0;𝜌+𝜎>0 (40) where 𝐻_𝑛(𝑥₁,𝑥₂) are the 2-variable Hermite-Kampé de-Fériet polynomials (2VHKdFP) [10].
Next, replacing 𝑥₁ by 𝑥, 𝑥_𝑟 by 𝑦 and taking 𝑥₂=𝑥₃=⋯=𝑥_𝑟−1=0 in (38) and using the relation 𝐻_𝑛^{(1,2,…,𝑟)}(𝑥,0,0,…,0,𝑦)=𝑔^𝑟_𝑛(𝑥,𝑦), we get ^𝑐_𝑎(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^{𝜇−𝜆−1}−𝑏()×exp𝑐−𝑎²()()𝑢−𝑎)(𝑐−𝑢+𝑥𝑡𝑢−𝑎^𝜌(𝑐−𝑢)^𝜎+𝑦𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎𝑟=𝑑𝑢_∞𝑛=0(𝑐−𝑎)^{𝜇+(𝜌+𝜎)𝑛−1}𝑔^𝑟_𝑛𝑡𝑥,𝑦×𝐵𝜆+𝜌𝑛,𝜇−𝜆+𝜎𝑛;𝑏^𝑛𝜇(𝜆)(𝑏),𝑛!Re>Re>0;Re>0;𝜌,𝜎≥0;𝜌+𝜎>0 (41) where 𝑔^𝑟_𝑛(𝑥,𝑦) are the Gould-Hopper generalized Hermite polynomials (GHGHP) [11].
Further, replacing 𝑥₁ by 𝜈𝑥 and taking 𝑥₂=𝑥₃=⋯=𝑥_𝑟−1=0, 𝑥_𝑟=−1 in (38) and using the relation 𝐻_𝑛^{(1,2,…,𝑟)}(𝜈𝑥,0,0,…,0,−1)=𝐻_{𝑛,𝑟,𝜈}(𝑥), we get ^𝑐_𝑎(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^{𝜇−𝜆−1}−𝑏()×exp𝑐−𝑎²()()𝑢−𝑎)(𝑐−𝑢+𝜈𝑥𝑡𝑢−𝑎^𝜌(𝑐−𝑢)^𝜎−𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎𝑟=𝑑𝑢_∞𝑛=0(𝑐−𝑎)^{𝜇+(𝜌+𝜎)𝑛−1}𝐻_{𝑛,𝑟,𝜈}(𝑥)𝑡×𝐵𝜆+𝜌𝑛,𝜇−𝜆+𝜎𝑛;𝑏^𝑛𝜇(𝜆)(𝑏),𝑛!Re>Re>0;Re>0;𝜌,𝜎≥0;𝜌+𝜎>0 (42) where 𝐻_{𝑛,𝑟,𝜈}(𝑥) are the generalized Hermite polynomials [12].
Furthermore, taking 𝑟=𝜈=2 in (42) and using the relation 𝐻_𝑛,2,2(𝑥)=𝐻_𝑛(𝑥), we get ^𝑐_𝑎(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^{𝜇−𝜆−1}−𝑏()×exp𝑐−𝑎²()()𝑢−𝑎)(𝑐−𝑢+2𝑥𝑡𝑢−𝑎^𝜌(𝑐−𝑢)^𝜎−𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎2=𝑑𝑢_∞𝑛=0(𝑐−𝑎)^{𝜇+(𝜌+𝜎)𝑛−1}𝐻_𝑛(𝑥)𝑡×𝐵𝜆+𝜌𝑛,𝜇−𝜆+𝜎𝑛;𝑏^𝑛𝜇(𝜆)(𝑏),𝑛!Re>Re>0;Re>0;𝜌,𝜎≥0;𝜌+𝜎>0 (43) where 𝐻_𝑛(𝑥) are the ordinary Hermite polynomials [1].
Case 2. Consider the generating function of the 2-variable Laguerre polynomials 𝐿_𝑛(𝑥,𝑦) [13] 11−𝑦𝑡exp−𝑥𝑡=1−𝑦𝑡^∞_𝑛=0𝐿_𝑛𝑥,𝑦𝑡^𝑛𝑦𝑡<1, (44) which can also be expressed as exp𝑦𝑡𝐶₀(𝑥𝑡)=^∞_𝑛=0𝐿_𝑛𝑡𝑥,𝑦^𝑛,𝑛! (45) where 𝐶₀(𝑥) denotes the 0th-order Tricomi function [4].
Applying Theorem 1 to generating functions (44) and (45), we find the following integrals: ^𝑐_𝑎(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^{𝜇−𝜆−1}1−𝑦𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎−𝑏()×exp𝑐−𝑎²()−()𝑢−𝑎)(𝑐−𝑢𝑥𝑡𝑢−𝑎^𝜌(𝑐−𝑢)^𝜎1−𝑦𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎=𝑑𝑢_∞𝑛=0(𝑐−𝑎)^{𝜇+(𝜌+𝜎)𝑛−1}𝐿_𝑛𝑡𝑥,𝑦×𝐵𝜆+𝜌𝑛,𝜇−𝜆+𝜎𝑛;𝑏^𝑛𝜇(𝜆)(𝑏)()Re>Re>0;Re>0;𝑦𝑡𝑢−𝑎^𝜌(𝑐−𝑢)^𝜎<1;𝜌,𝜎≥0;𝜌+𝜎>0_𝑐𝑎,(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^{𝜇−𝜆−1}−𝑏()×exp𝑐−𝑎²()()𝑢−𝑎)(𝑐−𝑢+𝑦𝑡𝑢−𝑎^𝜌(𝑐−𝑢)^𝜎×𝐶₀𝑥𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎=𝑑𝑢_∞𝑛=0(𝑐−𝑎)^{𝜇+(𝜌+𝜎)𝑛−1}𝐿_𝑛𝑡𝑥,𝑦×𝐵𝜆+𝜌𝑛,𝜇−𝜆+𝜎𝑛;𝑏^𝑛𝜇(𝜆)(𝑏),𝑛!Re>Re>0;Re>0;𝜌,𝜎≥0;𝜌+𝜎>0 (46)(47) respectively.
Taking 𝑦=1 in (46) and (47) and using the relation 𝐿_𝑛(𝑥,1)=𝐿_𝑛(𝑥), we get ^𝑐_𝑎(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^{𝜇−𝜆−1}1−𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎−𝑏()×exp𝑐−𝑎²()−()𝑢−𝑎)(𝑐−𝑢𝑥𝑡𝑢−𝑎^𝜌(𝑐−𝑢)^𝜎1−𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎=𝑑𝑢_∞𝑛=0(𝑐−𝑎)^{𝜇+(𝜌+𝜎)𝑛−1}𝐿_𝑛(𝑥)×𝐵𝜆+𝜌𝑛,𝜇−𝜆+𝜎𝑛;𝑏𝑡^𝑛𝜇(𝜆)(𝑏)𝑡()Re>Re>0;Re>0;𝑢−𝑎^𝜌(𝑐−𝑢)^𝜎<1;𝜌,𝜎≥0;𝜌+𝜎>0_𝑐𝑎,(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^{𝜇−𝜆−1}−𝑏()×exp𝑐−𝑎²()()𝑢−𝑎)(𝑐−𝑢+𝑡𝑢−𝑎^𝜌(𝑐−𝑢)^𝜎×𝐶₀𝑥𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎=𝑑𝑢_∞𝑛=0(𝑐−𝑎)^{𝜇+(𝜌+𝜎)𝑛−1}𝐿_𝑛(𝑥)𝑡×𝐵𝜆+𝜌𝑛,𝜇−𝜆+𝜎𝑛;𝑏^𝑛𝜇(𝜆)(𝑏),𝑛!Re>Re>0;Re>0;𝜌,𝜎≥0;𝜌+𝜎>0 (48)(49) respectively, where 𝐿_𝑛(𝑥) are the ordinary Laguerre polynomials [1].
Again, taking 𝑦=0 in (46) and (47) and using the relation 𝐿_𝑛(𝑥,0)=(−𝑥)^𝑛/𝑛!, we get ^𝑐_𝑎(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^{𝜇−𝜆−1}−𝑏()×exp𝑐−𝑎²()()𝑢−𝑎)(𝑐−𝑢−𝑥𝑡𝑢−𝑎^𝜌(𝑐−𝑢)^𝜎=𝑑𝑢_∞𝑛=0(𝑐−𝑎)^{𝜇+(𝜌+𝜎)𝑛−1}()×𝐵𝜆+𝜌𝑛,𝜇−𝜆+𝜎𝑛;𝑏−𝑥𝑡^𝑛𝜇(𝜆)(𝑏),𝑛!Re>Re>0;Re>0;𝜌,𝜎≥0;𝜌+𝜎>0_𝑐𝑎(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^{𝜇−𝜆−1}−𝑏()×exp𝑐−𝑎²()𝐶𝑢−𝑎)(𝑐−𝑢⁰𝑥𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎=𝑑𝑢_∞𝑛=0(𝑐−𝑎)^{𝜇+(𝜌+𝜎)𝑛−1}()×𝐵𝜆+𝜌𝑛,𝜇−𝜆+𝜎𝑛;𝑏−𝑥𝑡^𝑛(𝑛!)²Re𝜇>Re(𝜆)>0;Re(𝑏)>0;𝜌,𝜎≥0;𝜌+𝜎>0, (50)(51) respectively.
Next, taking 𝑥=0 in (46) and (47) and using the relation 𝐿_𝑛(0,𝑦)=𝑦^𝑛, we get ^𝑐_𝑎(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^{𝜇−𝜆−1}1−𝑦𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎−𝑏()exp𝑐−𝑎²()=𝑢−𝑎)(𝑐−𝑢𝑑𝑢_∞𝑛=0(𝑐−𝑎)^{𝜇+(𝜌+𝜎)𝑛−1}×𝐵𝜆+𝜌𝑛,𝜇−𝜆+𝜎𝑛;𝑏𝑦𝑡^𝑛Re𝜇>Re(𝜆)>0;Re(𝑏)>0;𝜌,𝜎≥0;𝜌+𝜎>0_𝑐𝑎,(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^{𝜇−𝜆−1}−𝑏()×exp𝑐−𝑎²()()𝑢−𝑎)(𝑐−𝑢+𝑦𝑡𝑢−𝑎^𝜌(𝑐−𝑢)^𝜎=^∞_𝑛=0(𝑐−𝑎)^{𝜇+(𝜌+𝜎)𝑛−1}×𝐵𝜆+𝜌𝑛,𝜇−𝜆+𝜎𝑛;𝑏𝑦𝑡^𝑛𝜇(𝜆)(𝑏),𝑛!Re>Re>0;Re>0;𝜌,𝜎≥0;𝜌+𝜎>0 (52)(53) respectively. Replacing 𝑦 by −𝑥 in (53), it reduces to (50).
Case 3. Consider the generating function of the Hermite-Appell polynomials _𝐻𝐴_𝑛(𝑥₁,𝑥₂,𝑥₃) [14, page 759 (2.3)] 𝐴(𝑡)exp𝑥₁𝑡+𝑥₂𝑡²+𝑥₃𝑡³=^∞_𝐻𝑛=0𝐴_𝑛𝑥₁,𝑥₂,𝑥₃𝑡^𝑛.𝑛! (54)
Applying Theorem 1 to generating function (54), we find the following integral: ^𝑐_𝑎(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^{𝜇−𝜆−1}𝐴𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎−𝑏()×exp𝑐−𝑎²()+𝑢−𝑎)(𝑐−𝑢³_𝑗=1𝑥_𝑗𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎𝑗=𝑑𝑢_∞𝑛=0(𝑐−𝑎)_𝐻^{𝜇+(𝜌+𝜎)𝑛−1}𝐴_𝑛𝑥₁,𝑥₂,𝑥₃𝑡×𝐵𝜆+𝜌𝑛,𝜇−𝜆+𝜎𝑛;𝑏^𝑛𝜇(𝜆)(𝑏).𝑛!Re>Re>0;Re>0;𝜌,𝜎≥0;𝜌+𝜎>0 (55)
Taking 𝑥₂=𝑥₃=0 and replacing 𝑥₁ by 𝑥 in (55) and using the relation _𝐻𝐴_𝑛(𝑥,0,0)=𝐴_𝑛(𝑥) [14, page 760 (2.6)], we get ^𝑐_𝑎(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^{𝜇−𝜆−1}𝐴𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎−𝑏()×exp𝑐−𝑎²()()𝑢−𝑎)(𝑐−𝑢+𝑥𝑡𝑢−𝑎^𝜌(𝑐−𝑢)^𝜎=𝑑𝑢_∞𝑛=0(𝑐−𝑎)^{𝜇+(𝜌+𝜎)𝑛−1}𝐴_𝑛(𝑥)𝑡×𝐵𝜆+𝜌𝑛,𝜇−𝜆+𝜎𝑛;𝑏^𝑛𝜇(𝜆)(𝑏),𝑛!Re>Re>0;Re>0;𝜌,𝜎≥0;𝜌+𝜎>0 (56) where 𝐴_𝑛(𝑥) are the Appell polynomials [4, 15].
Case 4. Consider the generating function of the Laguerre-Appell polynomials _𝐿𝐴_𝑛(𝑥,𝑦) [16] 𝐴(𝑡1)1−𝑦𝑡exp−𝑥𝑡=1−𝑦𝑡_{∞𝐿𝑛=0}𝐴_𝑛𝑥,𝑦𝑡^𝑛𝑦𝑡<1, (57) which can also be expressed as follows: 𝐴(𝑡)exp𝑦𝑡𝐶₀(𝑥𝑡)=^∞_𝐿𝑛=0𝐴_𝑛𝑡𝑥,𝑦^𝑛.𝑛! (58)
Applying Theorem 1 to generating functions (57) and (58), we find the following integrals: ^𝑐_𝑎(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^{𝜇−𝜆−1}𝐴𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎1−𝑦𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎−𝑏()×exp𝑐−𝑎²()−()𝑢−𝑎)(𝑐−𝑢𝑥𝑡𝑢−𝑎^𝜌(𝑐−𝑢)^𝜎1−𝑦𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎=𝑑𝑢_∞𝑛=0(𝑐−𝑎)_𝐿^{𝜇+(𝜌+𝜎)𝑛−1}𝐴_𝑛𝑡𝑥,𝑦×𝐵𝜆+𝜌𝑛,𝜇−𝜆+𝜎𝑛;𝑏^𝑛𝜇(𝜆)(𝑏)()Re>Re>0;Re>0;𝑦𝑡𝑢−𝑎^𝜌(𝑐−𝑢)^𝜎<1;𝜌,𝜎≥0;𝜌+𝜎>0_𝑐𝑎,(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^{𝜇−𝜆−1}𝐴𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎−𝑏()×exp𝑐−𝑎²()()𝑢−𝑎)(𝑐−𝑢+𝑦𝑡𝑢−𝑎^𝜌(𝑐−𝑢)^𝜎×𝐶₀𝑥𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎=𝑑𝑢_∞𝑛=0(𝑐−𝑎)_𝐿^{𝜇+(𝜌+𝜎)𝑛−1}𝐴_𝑛𝑡𝑥,𝑦×𝐵𝜆+𝜌𝑛,𝜇−𝜆+𝜎𝑛;𝑏^𝑛𝜇(𝜆)(𝑏),𝑛!Re>Re>0;Re>0;𝜌,𝜎≥0;𝜌+𝜎>0 (59)(60) respectively.
Taking 𝑦=0 and replacing 𝑥 by −𝑥 in (59) and (60) and using the relation _𝐿𝐴_𝑛(𝑥,0)=𝐴_𝑛(−𝑥)/𝑛! [16, page 9 (2.8)], we get (56) and ^𝑐_𝑎(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^{𝜇−𝜆−1}𝐴𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎−𝑏()×exp𝑐−𝑎²()𝐶𝑢−𝑎)(𝑐−𝑢⁰−𝑥𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎=𝑑𝑢_∞𝑛=0(𝑐−𝑎)^{𝜇+(𝜌+𝜎)𝑛−1}𝐴_𝑛(𝑥)𝑡×𝐵𝜆+𝜌𝑛,𝜇−𝜆+𝜎𝑛;𝑏^𝑛(𝑛!)²Re𝜇>Re(𝜆)>0;Re(𝑏)>0;𝜌,𝜎≥0;𝜌+𝜎>0, (61) respectively.
Case 5. Consider the generating function of the Hermite-Sheffer polynomials _𝐻𝑠_𝑛(𝑥₁,𝑥₂,𝑥₃) [17] 𝐴(𝑡)exp𝑥₁𝐻(𝑡)+𝑥₂𝐻²(𝑡)+𝑥₃𝐻³(𝑡=_{∞𝐻𝑛=0})𝑠_𝑛𝑥₁,𝑥₂,𝑥₃𝑡^𝑛.𝑛! (62)
Applying Theorem 1 to generating function (62), we find the following integral: ^𝑐_𝑎(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^{𝜇−𝜆−1}𝐴𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎−𝑏()×exp𝑐−𝑎²()+𝑢−𝑎)(𝑐−𝑢_3𝑗=1𝑥_𝑗𝐻^𝑗𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎=𝑑𝑢_∞𝑛=0(𝑐−𝑎)_𝐻^{𝜇+(𝜌+𝜎)𝑛−1}𝑠_𝑛𝑥₁,𝑥₂,𝑥₃𝑡×𝐵𝜆+𝜌𝑛,𝜇−𝜆+𝜎𝑛;𝑏^𝑛𝜇(𝜆)(𝑏).𝑛!Re>Re>0;Re>0;𝜌,𝜎≥0;𝜌+𝜎>0 (63)
Taking 𝑥₂=𝑥₃=0 and replacing 𝑥₁ by 𝑥 in (63) and using the relation _𝐻𝑠_𝑛(𝑥,0,0)=𝑠_𝑛(𝑥) [17, page 16 (2.3)], we get ^𝑐_𝑎(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^{𝜇−𝜆−1}𝐴𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎−𝑏()×exp𝑐−𝑎²()𝑡()𝑢−𝑎)(𝑐−𝑢+𝑥𝐻𝑢−𝑎^𝜌(𝑐−𝑢)^𝜎=𝑑𝑢_∞𝑛=0(𝑐−𝑎)^{𝜇+(𝜌+𝜎)𝑛−1}𝑠_𝑛(𝑥)𝑡×𝐵𝜆+𝜌𝑛,𝜇−𝜆+𝜎𝑛;𝑏^𝑛𝜇(𝜆)(𝑏),𝑛!Re>Re>0;Re>0;𝜌,𝜎≥0;𝜌+𝜎>0 (64) where 𝑠_𝑛(𝑥) are the Sheffer polynomials [15, 18, 19].

4. Concluding Remarks

In this paper, we have derived the evaluations of certain Euler type integrals. We have also established a theorem and applied it to obtain evaluations of some integrals in terms of EBF 𝐵(𝛼,𝛽;𝑏). We remark that these results can be extended to multivariable case.

To give an example, we consider the following integral representation [5, page 965 (20)] of Lauricella’s multiple hypergeometric series 𝐹_𝐷^(𝑛) [4]: ¹₀𝑡^𝛼−1(1−𝑡)^𝑛𝛽−1_𝑖=11−𝑥_𝑖𝑡^−𝛼𝑖𝐹𝑑𝑡=𝐵𝛼,𝛽_𝐷^(𝑛)𝛼,𝛼₁,𝛼₂,…,𝛼_𝑛;𝛼+𝛽;𝑥₁,𝑥₂,…,𝑥_𝑛Re(𝛼),Re𝛽>0;max𝑥₁,𝑥₂,…,𝑥_𝑛<1. (65)

Now, taking 𝜙(𝑡)=∏^𝑛_𝑖=1(1−𝑥_𝑖𝑡)^−𝛼𝑖,𝜓(𝑡)=1/𝑡(1−𝑡), 𝑎=0, and 𝑒=𝛾=1 in (15), using integral (65) in the r.h.s. after expanding _𝑟𝐹_𝑡^{(𝑓𝑟);𝑑𝜓(𝑡)}_(𝑔𝑡); in series and simplifying, we get𝐼_{𝑑,𝛼,𝛽,1,0,1}^𝑛_𝑖=11−𝑥_𝑖𝑡^−𝛼𝑖,1𝑡()=1−𝑡_{∞𝑚1,𝑚2,…,𝑚𝑛=0}(𝛼)_{𝑚1+𝑚2+⋯+𝑚𝑛}𝛼_1𝑚1𝛼_2𝑚2⋯𝛼_𝑛𝑚𝑛𝛼+𝛽_{𝑚1+𝑚2+⋯+𝑚𝑛}𝑥^𝑚1₁𝑚₁!𝑥^𝑚2₂𝑚₂!⋯𝑥^𝑚𝑛_𝑛𝑚_𝑛!×_𝑟+2𝐹_𝑡+2𝑓_𝑟,121−𝛼−𝛽−^𝑛_𝑖=1𝑚_𝑖1,1−2𝛼+𝛽+^𝑛_𝑖=1𝑚_𝑖;𝑔4𝑑_𝑡,1−𝛼+^𝑛_𝑖=1𝑚_𝑖(𝛼)𝛽𝑥,1−𝛽;Re,Re>0;max₁,𝑥₂,…,𝑥_𝑛<1. (66)

We note that, for 𝛼₃=𝛼₄=⋯=0, (66) reduces to (17).

Further, we extend Theorem 1 as follows.

Theorem 5. Let the conditions for 𝐺(𝑥₁,𝑥₂,…,𝑥_𝑟;𝑡) defined by (31) be the same as in Theorem 1, and then one has ^𝑐_𝑎(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^{𝜇−𝜆−1}×^𝑠_𝑖=11−𝑦_𝑖(𝑢−𝑎)^−𝛼𝑖×𝐺𝑥₁,𝑥₂,…,𝑥_𝑟;𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎−𝑏()×exp𝑐−𝑎²()=𝑢−𝑎)(𝑐−𝑢𝑑𝑢_{∞𝑛,𝑚1,𝑚2,…,𝑚𝑠=0}(𝑐−𝑎)^{𝜇+(𝜌+𝜎)𝑛+𝑚1+𝑚2+⋯+𝑚𝑠−1}×𝑐_𝑛𝑔_𝑛𝑥₁,𝑥₂,…,𝑥_𝑟×𝐵𝜆+𝜌𝑛+^𝑠_𝑖=1𝑚_𝑖,𝜇−𝜆+𝜎𝑛;𝑏^𝑠_𝑖=1𝛼_𝑖𝑚𝑖𝑦^𝑚𝑖_𝑖𝑚_𝑖!𝑡^𝑛Re𝜇>Re(𝜆)>0;Re(𝑏)>0;𝜌,𝜎≥0;𝜌+𝜎>0. (67)
The proof of this theorem is similar to that of Theorem 1.
The following are the consequences of Theorem 5.

Corollary 6. Let the conditions for 𝐺(𝑥₁,𝑥₂,…,𝑥_𝑟;𝑡) defined by (31) be the same as in Theorem 1, and then for 𝑏=0 and 𝑠=2, one has ^𝑐_𝑎(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^{𝜇−𝜆−1}ײ_𝑖=11−𝑦_𝑖(𝑢−𝑎)^−𝛼𝑖×𝐺𝑥₁,𝑥₂,…,𝑥_𝑟;𝑡(𝑢−𝑎)^𝜌(𝑐−𝑢)^𝜎=𝑑𝑢_∞𝑛=0(𝑐−𝑎)^{𝜇+(𝜌+𝜎)𝑛−1}𝑐_𝑛𝑔_𝑛𝑥₁,𝑥₂,…,𝑥_𝑟×𝐵𝜆+𝜌𝑛,𝜇−𝜆+𝜎𝑛×𝐹₁𝜆+𝜌𝑛,𝛼₁,𝛼₂()𝑦;𝜇+𝜌+𝜎𝑛;𝑐−𝑎₁,(𝑐−𝑎)𝑦₂𝑡^𝑛Re𝜇>Re(𝜆)>0;𝜌,𝜎≥0;𝜌+𝜎>0. (68)

Corollary 7. Let the conditions for 𝐺(𝑥₁,𝑥₂,…,𝑥_𝑟;𝑡) defined by (31) be the same as in Theorem 1, and then for 𝑏=0 and 𝜌=𝜎=1, one has ^𝑐_𝑎(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^{𝜇−𝜆−1}×^𝑠_𝑖=11−𝑦_𝑖(𝑢−𝑎)^−𝛼𝑖×𝐺𝑥₁,𝑥₂,…,𝑥_𝑟;𝑡(𝑢−𝑎)(𝑐−𝑢)𝑑𝑢=𝐵𝜆,𝜇−𝜆^∞_𝑛=0(𝑐−𝑎)^{𝜇+2𝑛−1}𝑐_𝑛𝑔_𝑛𝑥₁,𝑥₂,…,𝑥_𝑟×(𝜆)_𝑛𝜇−𝜆_𝑛𝜇_2𝑛×𝐹_𝐷^(𝑠)𝜆+𝑛,𝛼₁,𝛼₂,…,𝛼_𝑠;𝜇+2𝑛;(𝑐−𝑎)𝑦₁,(𝑐−𝑎)𝑦₂,…,(𝑐−𝑎)𝑦_𝑠𝑡^𝑛Re𝜇>Re(𝜆)>0. (69)

Example 8. Applying Corollary 7 to generating function (37) of the multivariable Hermite polynomials 𝐻_𝑛^{(1,2,…,𝑟)}(𝑥₁,𝑥₂,…,𝑥_𝑟), we get ^𝑐_𝑎(𝑢−𝑎)^𝜆−1(𝑐−𝑢)^{𝜇−𝜆−1}×^𝑠_𝑖=11−𝑦_𝑖(𝑢−𝑎)^−𝛼𝑖×exp^𝑟_𝑗=1𝑥_𝑗(𝑡(𝑢−𝑎)(𝑐−𝑢))^𝑗𝑑𝑢=𝐵𝜆,𝜇−𝜆^∞_𝑛=0(𝑐−𝑎)^{𝜇+2𝑛−1}𝐻_𝑛^{(1,2,…,𝑟)}𝑥₁,𝑥₂,…,𝑥_𝑟×(𝜆)_𝑛𝜇−𝜆_𝑛𝜇_2𝑛×𝐹_𝐷^(𝑠)𝜆+𝑛,𝛼₁,𝛼₂,…,𝛼_𝑠;𝜇+2𝑛;(𝑐−𝑎)𝑦₁,(𝑐−𝑎)𝑦₂,…,(𝑐−𝑎)𝑦_𝑠𝑡^𝑛𝜇(𝜆).𝑛!Re>Re>0 (70)

Remark 9. Taking 𝑎=0 and 𝑐=1 in (67), we get an extension of integral (34).

Remark 10. Taking 𝑎=−1 and 𝑐=1 in (67), we get an extension of integral (35).

Clearly, it appears that, by applying Theorem 5 and Corollaries 6 and 7 to the generating functions defined by (37), (44), (45), (54), (57), (58), and (62) and also to some other generating functions, a number of interesting integrals can be obtained.

Further, it is remarked that the operational methods provide useful tools to estimate specific theorems in various fields of analysis. This approach requires the validation of any of its consequences by a more rigorous procedure. This is indeed the case of the Ramanujan master theorem [20, 21]. An operational method, already employed to formulate a generalization of the Ramanujan master theorem, is applied for the evaluation of certain integrals by Babusci et al. [22].

The technique adopted in [22] provides a very flexible and powerful tool yielding new results encompassing different aspects of the theory of special functions. The possibility of using the method outlined in [22], for the integrals evaluated in this article, is a further research problem.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to express their thanks to the reviewer(s) for helpful suggestions and comments towards the improvement of this paper.