# A Note on Certain Laplace Transforms of Convolution-Type Integrals Involving Product of Two Generalized Hypergeometric Functions

Authors: Arjun Kumar Rathie; Young Hee Geum; Hwajoon Kim
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The aim of this research paper is to provide as many as forty-five attractive Laplace transforms of convolution type related to the product of generalized hypergeometric functions. These are achieved by employing summation theorems for the seriespFp−1 (for p=2,3,4, and 5) available in the literature. The obtained research result is to provide an easier method than the existing method.


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## Body

## 1. Introduction and Results’ Required

The theory of hypergeometric and generalized hypergeometric functions [1–3] are fundamental in the field of mathematics, engineering mathematics, and mathematical physics. Most of the commonly used functions that occur in the analysis are special cases or limiting cases of F12 and F11.It is well known thatF12 and F11 defined by(1)F12a,bc|z=∑n=0∞anbncnznn!,F11ac|z=∑n=0∞ancnznn!,where an=Γa+n/Γa and a≠0. Applications related to the detailed content can be found in [4–7].A natural generalization of this function can be represented by(2)Fqpab|z=Fqpa1,…,apb1,…,bq|z=∑n=0∞a1n⋯apnb1n⋯bqnznn!.In the theory of hypergeometric and generalized hypergeometric functions, the following classical summation theorems for the seriesFp−1p (for p=2,3,4, and 5) play an important role [3, 4].Gauss’ summation theorem [8]:(3)F12a,bc|1=ΓcΓc−a−bΓc−aΓc−b=Δ1a,b,c,which provided ℜc−a−b>0.Gauss’s second summation theorem [9]:(4)F12a,b12a+b+1|12=Γ1/2Γ1/2a+b+1Γ1/2a+1/2Γ1/2b+1/2=Δ2a,b.Bailey’s summation theorem [8]:(5)F12a,1−ab|12=Γ1/2bΓ1/2b+1/2Γ1/2b+1/2aΓ1/2b−1/2a+1/2=Δ3a,b.Kummer’s summation theorem [8]:(6)F12a,b1+a−b|−1=Γ1+1/2aΓ1+a−bΓ1+aΓ1+1/2a−b=Δ4a,b.Watson summation theorem [8]:(7)F23a,b,c1/2a+b+1,2c|1=Γ1/2Γc+1/2Γ1/2a+1/2b+1/2Γc−1/2a−1/2b+1/2Γ1/2a+1/2Γ1/2b+1/2Γc−1/2a+1/2Γc−1/2b+1/2=Δ5a,b,c,which provided ℜ2c−a−b>−1.Dixon’s summation theorem [9]:(8)F23a,b,c1+a−b,1+a−c|1=Γ1+1/2aΓ1+a−bΓ1+a−cΓ1+1/2a−b−cΓ1+aΓ1+1/2a−bΓ1+1/2a−cΓ1+a−b−c=Δ6a,b,c,which provided ℜa−2b−2c>−2.Whipple’s summation theorem [9]:(9)F23a,1−a,ce,1+2c−e|1=21−2cπΓeΓ1+2c−eΓ1/2a+1/2eΓ1/2−1/2a+1/2eΓc+1/2a−1/2e+1/2Γc−1/2a−1/2e+1=Δ7a,c,e,which provided ℜc>0.Second Whipple’s summation theorem [1]:(10)F34a,1+12a,b,c12a,a−b+1,a−c+1|−1=Γa−b+1Γa−c+1Γa+1Γa−b−c+1=Δ8a,b,c.Dougall’s summation theorem [1]:(11)F45a,1+12a,c,d,e12a,a−c+1,a−d+1,a−e+1|1=Γa−c+1Γa−d+1Γa−e+1Γa−c−d−e+1Γa+1Γa−d−e+1Γa−c−e+1Γa−c−d+1=Δ9a,c,d,e.In addition to this, we have the following general result of the Laplace transform of convolution-type integrals involving the product of two generalized hypergeometric functions available in the literature, see [6, 7]:(12)∫0∞e−st∫0tτμ−1t−τν−1Fqpab|kτp′Fq′a′b′|k′t−τdτdt=ΓμΓνs−μ−νFqp+1a,μb|ksp′+1Fq′a′,νb′|k′s.The aim of this note is to demonstrate how one can easily obtain in all forty-five Laplace transforms of convolution type related to the product of two generalized hypergeometric functions from the general result (12) by employing various summation theorems (3) to (11). The results obtained earlier by Milovanovic̃ et al. [1, 2] follow special cases of our main findings.


## 2. A Note on Certain Laplace Transforms of Convolution-Type Integrals Involving Product of Two Generalized Hypergeometric Functions

In this section, we shall establish in all forty-five Laplace transforms of convolution type related to the product of two generalized hypergeometric functions mentioned in the following theorems. All delta values that appear in this section are shown in Section1.Theorem 1.
Forℜs>0, ℜd>0, ℜd′>0, ℜc−a−b>0, and ℜc′−a′−b′>0, the following result holds true:(13)∫0∞e−st∫0tτd−1t−τd′−1F22a,bc,d|τsF22a′,b′c′,d′|t−τsdτdt=ΓdΓd′s−d−d′Δ1a,b,cΔ1a′,b′,c′.Theorem 2.
Forℜs>0, ℜd>0, ℜd′>0, and ℜc−a−b>0, the following result holds true:(14)∫0∞e−st∫0tτd−1t−τd′−1F22a,bc,d|τs×F22a′,b′12a′+b′+1,d′|12t−τsdτdt=ΓdΓd′s−b−b′Δ1a,b,cΔ2a′,b′.Theorem 3.
Forℜs>0, ℜd>0, ℜd′>0, and ℜc−a−b>0, the following result holds true:(15)∫0∞e−st∫0tτd−1t−τd′−1F22a,bc,d|τs×F22a′,1−a′b′,d′|12t−τsdτdt=ΓdΓd′s−d−d′Δ1a,b,cΔ3a′,b′.Theorem 4.
Forℜs>0, ℜd>0, ℜd′>0, and ℜc−a−b>0, the following result holds true:(16)∫0∞e−st∫0tτd−1t−τd′−1F22a,bc,d|τs×F22a′,b′1+a′−b′,d′|t−τsdτdt=ΓdΓd′s−d−d′Δ1a,b,cΔ4a′,b′.Theorem 5.
Forℜs>0, ℜd>0, ℜd′>0, ℜc−a−b>0, and ℜ2c′−a′−b′>−1, the following result holds true:(17)∫0∞e−st∫0tτd−1t−τd′−1F22a,bc,d|τs×F33a′,b′,c′12a′+b′+1,2c′,d′|t−τsdτdt=ΓdΓd′sd+d′Δ1a,b,cΔ5a′,b′,c′.Theorem 6.
Forℜs>0, ℜd>0, ℜd′>0, ℜc−a−b>0, and ℜa′−2b′−2c′>−2, the following result holds true:(18)∫0∞e−st∫0tτd−1t−τd′−1F22a,bc,d|τs×F33a′,b′,c′1+a′−b′,1+a′−c′,d′|t−τsdτdt=ΓdΓd′sd+d′Δ1a,b,cΔ6a′,b′,c′.Theorem 7.
Forℜs>0, ℜd>0, ℜd′>0, and ℜc−a−b>0, the following result holds true:(19)∫0∞e−st∫0tτd−1t−τd′−1F22a,bc,d|τs×F33a′,1−a′,c′e′,1+2c′−e′,d′|t−τsdτdt=ΓdΓd′sd+d′Δ1a,b,cΔ7a′,c′,e′.Theorem 8.
Forℜs>0, ℜd>0, ℜd′>0, and ℜc−a−b>0, the following result holds true:(20)∫0∞e−st∫0tτd−1t−τd′−1F22a,bc,d|τs×F44a′,1+12a′,b′,c′12a′,a′−b′+1,a′−c′+1,d′|−t−τsdτdt=ΓdΓd′sd+d′Δ1a,b,cΔ8a′,b′,c′.Theorem 9.
Forℜs>0, ℜd>0, ℜf′>0, ℜc−a−b>0, and ℜa′−c′−d′−e′+1>0, the following result holds true:(21)∫0∞e−st∫0tτd−1t−τf′−1F22a,bc,d|τs×F55a′,1+12a′,c′,d′,e′12a′,a′−c′+1,a′−d′+1,a′−e′+1,f′|t−τsdτdt=ΓdΓf′sd+f′Δ1a,b,cΔ9a′,c′,d′,e′.Theorem 10.
Forℜs>0, ℜd>0, and ℜd′>0, the following result holds true:(22)∫0∞e−st∫0tτd−1t−τd′−1F22a,b12a+b+1,d|12τs×F22a′,b′12a′+b′+1,d′|12t−τsdτdt=ΓdΓd′sd+d′Δ2a,bΔ2a′,b′.Theorem 11.
Forℜs>0, ℜd>0, and ℜd′>0, the following result holds true:(23)∫0∞e−st∫0tτd−1t−τd′−1F22a,b12a+b+1,d|12τs×F22a′,1−a′b′,d′|12t−τsdτdt=ΓdΓd′sd+d′Δ2a,bΔ3a′,b′.Theorem 12.
Forℜs>0, ℜd>0, and ℜd′>0, the following result holds true:(24)∫0∞e−st∫0tτd−1t−τd′−1F22a,b12a+b+1,d|12τs×F22a′,b′1+a′−b′,d′|−t−τsdτdt=ΓdΓd′sd+d′Δ2a,bΔ4a′,b′.Theorem 13.
Forℜs>0, ℜd>0, ℜd′>0, and ℜ2c′−a′−b′>−1, the following result holds true:(25)∫0∞e−st∫0tτd−1t−τd′−1F22a,b12a+b+1,d|12τs×F33a′,b′,c′12a′+b′+1,2c′,d′|t−τsdτdt=ΓdΓd′sd+d′Δ2a,bΔ5a′,b′,c′.Theorem 14.
Forℜs>0, ℜd>0, ℜd′>0, and ℜa′−2b′−2c′>−2, the following result holds true:(26)∫0∞e−st∫0tτd−1t−τd′−1F22a,b12a+b+1,d|12τs×F33a′,b′,c′1+a′−b′,1+a′−c′,d′|t−τsdτdt=ΓdΓd′sd+d′Δ2a,bΔ6a′,b′,c′.Theorem 15.
Forℜs>0, ℜd>0, and ℜd′>0, the following result holds true:(27)∫0∞e−st∫0tτd−1t−τd′−1F22a,b12a+b+1,d|12τs×F33a′,1−a′,c′e′,1+2c′−e′,d′|t−τsdτdt=ΓdΓd′sd+d′Δ2a,bΔ7a′,c′,e′.Theorem 16.
Forℜs>0, ℜd>0, and ℜd′>0, the following result holds true:(28)∫0∞e−st∫0tτd−1t−τd′−1F22a,b12a+b+1,d|12τs×F44a′,1+12a′,b′,c′12a′,a′−b′+1,a′−c′+1,d′|−t−τsdτdt=ΓdΓd′sd+d′Δ2a,bΔ8a′,b′,c′.Theorem 17.
Forℜs>0, ℜd>0, ℜf′>0, and ℜa′−c′−d′−e′+1>0, the following result holds true:(29)∫0∞e−st∫0tτd−1t−τf′−1F22a,b12a+b+1,d|12τs×F55a′,1+12a′,c′,d′,e′12a′,a′−c′+1,a′−d′+1,a′−e′+1,f′|t−τsdτdt=ΓdΓf′sd+f′Δ2a,bΔ9a′,c′,d′,e′.Theorem 18.
Forℜs>0, ℜd>0, and ℜd′>0, the following result holds true:(30)∫0∞e−st∫0tτd−1t−τd′−1F22a,1−ab,d|12τs×F22a′,1−a′b′,d′|12t−τsdτdt=ΓdΓd′sd+d′Δ3a,bΔ3a′,b′.Theorem 19.
Forℜs>0, ℜd>0, and ℜd′>0, the following result holds true:(31)∫0∞e−st∫0tτd−1t−τd′−1F22a,1−ab,d|12τs×F22a′,b′1+a′−b′,d′|−t−τsdτdt=ΓdΓd′sd+d′Δ3a,bΔ4a′,b′.Theorem 20.
Forℜs>0, ℜd>0, ℜd′>0, and ℜ2c′−a′−b′>−1, the following result holds true:(32)∫0∞e−st∫0tτd−1t−τd′−1F22a,1−ab,d|12τs×F33a′,b′,c′12a′+b′+1,2c′,d′|t−τsdτdt=ΓdΓd′sd+d′Δ3a,bΔ5a′,b′,c′.Theorem 21.
Forℜs>0, ℜd>0, ℜd′>0, and ℜa′−2b′−c′>−2, the following result holds true:(33)∫0∞e−st∫0tτd−1t−τd′−1F22a,1−ab,d|12τs×F33a′,b′,c′1+a′−b′,1+a′−c′,d′|t−τsdτdt=ΓdΓd′sd+d′Δ3a,bΔ6a′,b′,c′.Theorem 22.
Forℜs>0, ℜd>0, and ℜd′>0, the following result holds true:(34)∫0∞e−st∫0tτd−1t−τd′F22a,1−ab,d|12τs×F33a′,1−a′,c′e′,1+2c′−e′,d′|t−τsdτdt=ΓdΓd′sd+d′Δ3a,bΔ7a′,c′,e′.Theorem 23.
Forℜs>0, ℜd>0, and ℜd′>0, the following result holds true:(35)∫0∞e−st∫0tτd−1t−τd′−1F22a,1−ab,d|12τs×F44a′,1+12a′,b′,c′12a′,a′−b′+1,a′−c′+1,d′|−t−τsdτdt=ΓdΓd′sd+d′Δ3a,bΔ8a′,b′,c′.Theorem 24.
Forℜs>0, ℜd>0, ℜf′>0, and ℜa′−c′−d′−e′+1>0, the following result holds true:(36)∫0∞e−st∫0tτd−1t−τf′−1F22a,1−ab,d|12τs×F55a′,1+12a′,c′,d′,e′12a′,a′−c′+1,a′−d′+1,a′−e′+1,f′|t−τsdτdt=ΓdΓf′sd+f′Δ3a,bΔ9a′,c′,d′,e′.Theorem 25.
Forℜs>0, ℜd>0, and ℜd′>0, the following result holds true:(37)∫0∞e−st∫0tτd−1t−τd′−1F22a,b1+a−b,d|−τs×F22a′,b′1+a′−b′,d′|−t−τsdτdt=ΓdΓd′sd+d′Δ4a,bΔ4a′,b′.Theorem 26.
Forℜs>0, ℜd>0, ℜd′>0, and ℜ2c′−a′−b′>−2, the following result holds true:(38)∫0∞e−st∫0tτd−1t−τd′−1F22a,b1+a−b,d|−τs×F33a′,b′,c′12a′+b′+1,2c′,d′|t−τsdτdt=ΓdΓd′sd+d′Δ4a,bΔ5a′,b′,c′.Theorem 27.
Forℜs>0, ℜd>0, ℜd′>0, and ℜa′−2b′−c′>−2, the following result holds true:(39)∫0∞e−st∫0tτd−1t−τd′−1F22a,b1+a−b,d|−τs×F33a′,b′,c′1+a′−b′,1+a′−c′,d′|t−τsdτdt=ΓdΓd′sd+d′Δ4a,bΔ6a′,b′,c′.Theorem 28.
Forℜs>0, ℜd>0, and ℜd′>0, the following result holds true:(40)∫0∞e−st∫0tτd−1t−τd′−1F22a,b1+a−b,d|−τs×F33a′,1−a′,c′e′,1+2c′−e′,d′|t−τsdτdt=ΓdΓd′sd+d′Δ4a,bΔ7a′,c′,e′.Theorem 29.
Forℜs>0, ℜd>0, and ℜd′>0, the following result holds true:(41)∫0∞e−st∫0tτd−1t−τd′−1F22a,b1+a−b,d|−τs×F44a′,1+12a′,b′,c′12a′,a′−b′+1,a′−c′+1,d′|−t−τsdτdt=ΓdΓd′sd+d′Δ4a,bΔ8a′,b′,c′.Theorem 30.
Forℜs>0, ℜd>0, ℜf′>0, and ℜa′−c′−d′−e′+1>0, the following result holds true:(42)∫0∞e−st∫0tτd−1t−τf′−1F22a,b1+a−b,d|−τs×F55a′,1+12a′,c′,d′,e′12a′,a′−c′+1,a′−d′+1,a′−e′+1,f′|t−τsdτdt=ΓdΓf′sd+f′Δ4a,bΔ9a′,c′,d′,e′.Theorem 31.
Forℜs>0, ℜd>0, ℜd′>0, ℜ2c−a−b>−1, and ℜ2c′−a′−b′>−1, the following result holds true:(43)∫0∞e−st∫0tτd−1t−τd′−1F33a,b,c12a+b+1,2c,d|τs×F33a′,b′,c′12a′+b′+1,2c′,d′|t−τsdτdt=ΓdΓd′sd+d′Δ5a,b,cΔ5a′,b′,c′.Theorem 32.
Forℜs>0, ℜd>0, ℜd′>0, ℜ2c−a−b>−1, and ℜa′−2b′−c′>−2, the following result holds true:(44)∫0∞e−st∫0tτd−1t−τd′−1F33a,b,c12a+b+1,2c,d|τs×F33a′,b′,c′1+a′−b′,1+a′−c′,d′|t−τsdτdt=ΓdΓd′sd+d′Δ5a,b,cΔ6a′,b′,c′.Theorem 33.
Forℜs>0, ℜd>0, ℜd′>0, and ℜ2c′−a′−b′>−1, the following result holds true:(45)∫0∞e−st∫0tτd−1t−τd′−1F33a,b,c12a+b+1,2c,d|τs×F33a′,1−a′,c′e′,1+2c′−e′,d′|t−τsdτdt=ΓdΓd′sd+d′Δ5a,b,cΔ7a′,c′,e′.Theorem 34.
Forℜs>0, ℜd>0, ℜd′>0, and ℜ2c−a−b>−1, the following result holds true:(46)∫0∞e−st∫0tτd−1t−τd′−1F33a,b,c12a+b+1,2c,d|τs×F44a′,1+12a′,b′,c′12a′,a′−b′+1,a′−c′+1,d′|−t−τsdτdt=ΓdΓd′sd+d′Δ5a,b,cΔ8a′,b′,c′.Theorem 35.
Forℜs>0, ℜd>0, ℜf′>0, ℜ2c−a−b>−1, and ℜa′−c′−d′−e′+1>0, the following result holds true:(47)∫0∞e−st∫0tτd−1t−τf′−1F33a,b12a+b+1,2c,d|τs×F55a′,1+12a′,c′,d′,e′12a′,a′−c′+1,a′−d′+1,a′−e′+1,f′|t−τsdτdt=ΓdΓf′sd+f′Δ5a,b,cΔ9a′,c′,d′,e′.Theorem 36.
Forℜs>0, ℜd>0, ℜd′>0, ℜa−2b−2c>−2, and ℜa′−2b′−c′>−2, the following result holds true:(48)∫0∞e−st∫0tτd−1t−τd′−1F33a,b,c1+a−b,1+a−c,d|τs×F33a′,b′,c′1+a′−b′,1+a′−c′,d′|t−τsdτdt=ΓdΓd′sd+d′Δ6a,b,cΔ6a′,b′,c′.Theorem 37.
Forℜs>0, ℜd>0, ℜd′>0, and ℜa−2b−2c>−2, the following result holds true:(49)∫0∞e−st∫0tτd−1t−τd′−1F33a,b,c1+a−b,1+a−c,d|τs×F33a′,1−a′,c′e′,1+2c′−e′,d′|t−τsdτdt=ΓdΓd′sd+d′Δ6a,b,cΔ7a′,c′,e′.Theorem 38.
Forℜs>0, ℜd>0, ℜd′>0, and ℜa−2b−2c>−2, the following result holds true:(50)∫0∞e−st∫0tτd−1t−τd′−1F33a,b,c1+a−b,1+a−c,d|τs×F44a′,1+12a′,b′,c′12a′,a′−b′+1,a′−c′+1,d′|−t−τsdτdt=ΓdΓd′sd+d′Δ6a,b,cΔ8a′,b′,c′.Theorem 39.
Forℜs>0, ℜd>0, ℜf′>0, ℜa−2b−2c>−2, and ℜa′−c′−d′−e′+1>0, the following result holds true:(51)∫0∞e−st∫0tτd−1t−τf′−1F33a,b,c1+a−b,1+a−c,d|τs×F55a′,1+12a′,c′,d′,e′12a′,a′−c′+1,a′−d′+1,a′−e′+1,f′|t−τsdτdt=ΓdΓf′sd+f′Δ6a,b,cΔ9a′,c′,d′,e′.Theorem 40.
Forℜs>0, ℜd>0, and ℜd′>0, the following result holds true:(52)∫0∞e−st∫0tτd−1t−τd′−1F33a,1−a,ce,1+2c−e,d|τs×F33a′,1−a′,c′e′,1+2c′−e′,d′|t−τsdτdt=ΓdΓd′sd+d′Δ7a,c,eΔ7a′,c′,e′.Theorem 41.
Forℜs>0, ℜd>0, and ℜd′>0, the following result holds true:(53)∫0∞e−st∫0tτd−1t−τd′−1F33a,1−a,ce,1+2c−e,d|τs×F44a′,1+12a′,b′,c′12a′,a′−b′+1,a′−c′+1,d′|−t−τsdτdt=ΓdΓd′sd+d′Δ7a,c,eΔ8a′,b′,c′.Theorem 42.
Forℜs>0, ℜd>0, ℜf′>0, and ℜa′−c′−d′−e′+1>0, the following result holds true:(54)∫0∞e−st∫0tτd−1t−τf′−1F33a,1−a,ce,1+2c−e,d|τs×F55a′,1+12a′,c′,d′,e′12a′,a′−c′+1,a′−d′+1,a′−e′+1,f′|t−τsdτdt=ΓdΓf′sd+f′Δ7a,c,eΔ9a′,c′,d′,e′.Theorem 43.
Forℜs>0, ℜd>0, and ℜd′>0, the following result holds true:(55)∫0∞e−st∫0tτd−1t−τd′−1F44a,1+12a,b,c12a,a−b+1,a−c+1,d|−τs×F44a′,1+12a′,b′,c′12a′,a′−b′+1,a′−c′+1,d′|−t−τsdτdt=ΓdΓd′sd+d′Δ8a,b,cΔ8a′,b′,c′.Theorem 44.
Forℜs>0, ℜd>0, ℜf′>0, and ℜa′−c′−d′−e′+1>0, the following result holds true:(56)∫0∞e−st∫0tτd−1t−τf′−1F44a,1+12a,b,c12a,a−b+1,a−c+1,d|−τs×F55a′,1+12a′,c′,d′,e′12a′,a′−c′+1,a′−d′+1,a′−e′+1,f′|t−τsdτdt=ΓdΓf′sd+f′Δ8a,c,eΔ9a′,c′,d′,e′.Theorem 45.
Forℜs>0, ℜf>0, ℜf′>0, ℜa−c−d−e+1>0, and ℜa′−c′−d′−e′+1>0, the following result holds true:(57)∫0∞e−st∫0tτf−1t−τf′−1F55a,1+12a,c,d,e12a,a−c+1,a−d+1,a−e+1,f|τs×F55a′,1+12a′,c′,d′,e′12a′,a′−c′+1,a′−d′+1,a′−e′+1,f′|t−τsdτdt=ΓfΓf′sf+f′Δ9a,c,d,eΔ9a′,c′,d′,e′.Proof.
The proofs of results (13) to (57) asserted in Theorems 1–45 are quite straight forward. For example, in order to prove result (13) asserted in Theorem 1, we proceed as follows. In the general result (9), if we get p=q=p′=q′=2, a1=a,a2=b,a1′=a′,a2′=b′,b1=c,b2=d, b1′=c′,b2′=d′,k=k′=s,μ=d, and ν=d′, then it takes the following form:(58)∫0∞e−st∫0tτd−1t−τd′−1F22a,bc,d|τsF22a′,b′c′,d′|t−τsdτdt=ΓdΓd′s−d−d′F12a,bc|1F12a′,b′c′|1.
We now observe that the twiceF12 appearing on the right-hand side of (58) can be calculated by Gauss’ summation theorem (3), and we easily arrive at the desired result (13). This completes the proof of result (13) asserted by Theorem 1. The remaining results (14)–(57) can be established in a similar way. We would like to leave the details to the interested readers.


## 3. Corollaries

In this section, we shall mention several known results of our main findings:(a)
In Theorems1, 2, 4, 40, 42, and 44, if we take d=b and d′=b′, we get the known results due to Milovanovic̃ et al. (Theorems 2.1, 2, and 2.4 in [6] and Corollary 1.3 and 1.5 in [7], respectively).(b)
In Theorems3 and 11, if we take d=b and d′=1−a′, we get the known results due to Milovanovic̃ et al. (Theorems 2.3 in [6] and Corollary 4 in [7], respectively).(c)
In Theorems5 to 8, 13–16, and 26–29, if we take d=b and d′=c′, we get the known results due to Milovanovic̃ et al. (Theorems 2.5 to 2.10, 2.14 to 2.16, 2.23, 2.25, and 2.29 in [6] respectively).(d)
In Theorems20–23, if we take d=1−a and d′=c′, we get the known results due to Milovanovic̃ et al. (Theorems 2.11 to 2.13 and 2.27 in [6], respectively).(e)
In Theorems31–34, 36–38, 40, 41, and 43, if we take d=c and d′=c′, we get the known results due to Milovanovic̃ et al. (Theorems 2.17 to 2.22, 2.31, 2.33, 2.35, and 2.37 [6], respectively).(f)
In Theorems9, 17, 30, 35, 39, 42, and 44, if we take d=b and f′=c′, we get the known results due to Milovanovic̃ et al. (Theorems 2.24, 2.26, 2.30, 2.32, 2.34, 2.36, and 2.38 in [6], respectively).(g)
In Theorem24, if we take d=1−a and f′=c, we get the known result due to Milovanovic̃ et al. (Theorem 2.28 in [6]).(h)
In Theorem45, if we take f=c and f′=c′, we get the known result due to Milovanovic̃ et al. (Theorem 2.39 in [6]).(i)
In Theorem18, if we take d=1−a and d′=1−a′, we get a known result due to Milovanovic̃ et al. (Corollary 2 in [7]).(j)
In Theorem19, if we take d=1−a and d′=b′, we get a known result due to Milovanovic̃ et al. (Corollary 6 in [7]).


## 4. Conclusion

In this note, our aim is to demonstrate how one can easily obtain in all forty-five attractive Laplace transforms of convolution type related to the generalized hypergeometric functions from the general result recorded in [5].Applications related to engineering will be addressed in the next study. It is hoped that the results could be of potential use in the areas of engineering mathematics and mathematical physics.



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