# Some Integrals Involving Meijer’s and Hypergeometric Functions Using the Coherent State Technique

Authors: Dusan Popov; Romeo Negrea
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this paper, we have examined a particular case of coherent states, defined so that their structure constants depend only on the products of the energy eigenvalues of the examined systems. In this manner, we have built all three kinds of coherent states (Barut–Girardello, Klauder–Perelomov, and Gazeau–Klauder). From the equation of the unitary operator decomposition, we have highlighted and used a so-called fundamental integral, to obtain some new integrals involving Meijer’s and hypergeometric functions. All calculations are made using the properties of the diagonal operator ordering technique. This is implicitly proved that the coherent state technique can be useful not only in different branches of physics (quantum mechanics, quantum optics, quantum information theory, and so on) but also in the deduction of new integrals involving generalized Meijer’s and hypergeometric functions. This approach can be considered as suitable “feedback” from physics to mathematics.


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## 1. Introduction

The special functions have been introduced and investigated extensively due mainly to their applications in diverse areas in mathematics, applied mathematics, physics, and engineering. Most of the special functions, which have various physical and technical applications, can be expressed in terms of generalized hypergeometric functions. Srivastava and Karlsson [1] introduced and analyzed the extension of the generalized hypergeometric functions utilizing the extended Pochhammer symbol. Other new extensions of Pochhammer symbol can be found in [2, 3]. Several interesting integral representations of the Euler type and Laplace type for some Gauss hypergeometric functions of three variables are given in [4]. Another extension of the generalized hypergeometric functions is defined in [5] where some of their properties are presented such as integral representations, derivative formulas, recurrence relations, and others [5, 6].The hypergeometric functions are used in a wide range of applications, such as integral representations, generating functions, recurrence relations, finite and infinite sums, analytic continuation, asymptotic behaviour, and in perturbation theory or in quantum theory [5]. Today, it is a fact that there exists a strong relation between hypergeometric functions and some special integrals which appear in different branches of physics (quantum mechanics, quantum optics, quantum information theory, and so on).In the last decades, the coherent state (CS) formalism, introduced almost a century ago by Schrödinger [7], as quasi-classical state, has proven to be a useful approach for several applications in quantum mechanics, quantum optics, and quantum information theory and practice. Over time, these CSs, originally formulated only for the one-dimensional quantum harmonic oscillator (HO-1D), have been extended to other quantum systems. Thus, CSs were defined in the following manner: Barut–Girardello CSs (BG-CSs) [8], Klauder–Perelomov CSs (KP-CSs) [9], and Gazeau–Klauder CSs (GK-CSs) [10].For the HO-1D, all three manners of defining the CSs lead to the same results, i.e., to the same expression of CSs. For this reason, these CSs are also namedcanonical or linear coherent states. The situation is different if we consider other quantum systems (named nonlinear): the three manners of building CSs lead to different expressions. Consequently, all these CSs are named nonlinear.Appl and Schiller [11] observed that all CSs (linear and nonlinear) are in fact particular cases of generalized coherent states, defined so that their normalizing function is a generalized hypergeometric function (G-HGF) pFqai1p;bj1q;z2, where p and q are positive numbers. The sets a1,a2,…,ap,≡ai1p and b1,b2,…,bq,≡bj1q are real (or complex) parameters.Decades after their introduction, CSs seemed to be neglected in terms of scientific interest. However, in the second half of the twentieth century, there was a tendency to “link” the definition of CSs to the theory of quantum groups associated with the system under examination [8, 9, 12–15]. Unfortunately, there are just a few quantum systems for which we are able to construct the associated group generators. The impact situation can be easily avoided if we consider a dimensionless Hamiltonian H^, having only the discrete energy spectrum with eigenvalues en:(1)H^n=enn.We suppose that these energy eigenvalues are ascendant and nondegenerate:e0=1〈e1〈e2〈…〈en.On the other hand, let us choose a pair of two Hermitian operators, the creationE^+ and the annihilation E^−, whose actions on the Fock vectors n are(2)E^−n=enn−1,E^+n=en+1n+1,E^+E^−n=enn.In other words, we haveH^=E^+E^−, and we point out here that the above pair of operators is not mandatory as the group generators of the quantum group attached to the examined quantum systems are not supersymmetric partners.On the other hand, Hongyi [16] introduced a new calculation technique, called the integration with ordered products (IWOP), applicable only to HO-1D, which greatly facilitates calculations and leads to a range of results, including new mathematical results. In a series of previous works, we have generalized the IWOP technique and applied it to the pair of creation E^+, and annihilation E^−, operators associated with any quantum system, linear or nonlinear. Thus, it was born the diagonal operator ordering technique (DOOT) was born, with which we obtained a series of useful results [17–19]. An additional result of using DOOT appears in this paper: we obtain a series of new integration relations involving Meijer's Gp,qm,nz2|… and generalized hypergeometric functions pFq…;…;z2. This is, in fact, the main purpose of this paper.The paper is organized as follows. In Section2, we present some primary elements, necessary to define coherent states, using elements of the diagonal operator ordering technique (DOOT). Section 3 is dedicated to defining the different types of coherent states (CSs) and revealing their main properties. Special attention is paid to the relationship of solving the unit operator, i.e., to its development (decomposition) with respect to the projectors of the coherent states. In Section 4, we use a DOOT rule, according to which, inside the sign # #, the normally ordered operators can be seen as simple numbers and, consequently, we can perform all algebraic operations according to the usual rules. Consequently, several general integrals are obtained in which the generalized hypergeometric functions participate. Their verification is done by customizing the indices of these functions, which leads to known results obtained by other methods. In Section 5, some observations are made in connection with the results obtained previously. Implicitly, the results show that the formalism of coherent states corroborated with the DOOT technique can be useful also for deducing new integrals from complex or real space.


## 2. Preliminaries

Generally, the energy eigenvaluesen are dependent of a set of real (or complex) parameters ai1p and bj1q. Examining the expression of energy eigenvalues, we can say that for each quantum system, there exist a specific and unique set of integers and positive numbers p and q, respectively, and a set of parameters ai1p and bj1q. Let us suppose that the dimensionless energy eigenvalues are nondegenerate and have the general expression (this assertion will be implicitly motivated by the following):(3)el=Cl∏j=1qbj−1+l∏i=1pai−1+l,l=1,2,3,…,where C is a dimensionless real constant.We use for their product the following notation:(4)ρp,qba|n≡∏l=1nel=Cnn!∏j=1qbjn∏i=1pain,where xn=Γx+n/Γx is the Pochhammer symbol and Γx is Euler’s gamma function.It is useful to write the positive constantsρp,qb/a|n (called “structure constants,” when referring to coherent states) also in the same manner:(5)ρp,qba|n=Γp,qabCnΓn+1∏j=1qΓbj+n∏i=1pΓai+n,Γp,qab≡∏i=1pΓai∏j=1qΓbj.The repeated action of the creation operatorE^+ on the vacuum (or ground) state 0 leads to the following result:(6)E^+n0=∏l=0nenn=ρp,qba|nn,(7)nn=1ρp,qb/a|nE^+n00E^+n.The completion relation (or the identity operator decomposition) for Fock vectors, if we assume that the Fock vector space can be finite or infinite dimensional, i.e., with dimensionnmax≡M≤∞, is(8)∑n=0Mnn=1,and then it becomes(9)∑n=0M1ρp,qb/a|nE^+n00E^−n=1.Due to a fruitful operator ordering technique—the diagonal operator ordering technique (DOOT) (see [17] (which is a generalization of the integration with ordered products (IWOP)) and [16]), the above relation can be rearranged as follows:(10)00∑n=0M1ρp,qb/a|n#E^+E^−n#=1.The above sum is just a generalized hypergeometric function (G-HGF), ifM⟶∞, or a generalized hypergeometric polynomial of the degree M (G-HGP), if M<∞, with the ordered operator product #E^+E^−# as argument, generally defined as(11)pFqMai1p;bj1q;1Cx=∑n=0M1ρp,qb/a|nxn=∑n=0M∏i=1pain∏j=1qbjnx/Cnn!.Consequently, the vacuum projector is(12)00=1#pFqMai1p;bj1q;E^+E^−/C#.Evidently, as we will see later, the vacuum projector is the same for all three kinds of CSs, i.e., it is independent of the definition of CSs.This G-HGF is characteristic for every quantum system. At the same time, the definition of G-HGF is the motivation for choosing the general expression of energy eigenvalues (3). Consequently, the function #pFqMai1p;bj1q;x/C# can be called “characteristic generalized hypergeometric functions or polynomials” or “associated hypergeometric functions or polynomials” of the quantum system.Generally, an arbitrary set of CSsz is labelled by a complex number z=zexpiϕ, z≤R≤∞, ϕ∈0,2π and can be developed in the Fock vector basis n as(13)z=1Nz2∑n=0Mznρp,qb/a|nn.All CSszmust satisfy some conditions, sometimes named Klauder’s minimal prescriptions[10]: the continuity of labelling, i.e., ifz′⟶z, then compulsoryz′−z⟶0; as well as the condition that CSs must be normalized to unity and nonorthogonal:(14)z|z′=1,forz′=z≠0,forz′≠z.Moreover, the most important condition is the so-calledcompletion relation or the resolution of the unity operator of the CS projectors:(15)∫dμzzz=1,with such an integration measure(16)dμz=d2zπhz=dϕ2πdz2hz,to ensure a positive weight function hz which must be determined separately, for each examined quantum system and each kind of CSs.To find the weight functionhz, we must have(17)∑n,n′=0Mnn′ρp,qb/a|nρp,qb/a|n′∫0Rdz2hzNz2∫02πdϕ2πz∗nz′n′=1.The normalization functionNz2 is obtained from the normalization condition of CSs z|z=1, so that(18)Nz2=pFqMai1p;bj1q;1Cz2.The angular integral isz2nδnn′, and, if we put h˜z≡hz/Nz2, we have to solve an integral moment problem (the so-called Stieltjes or Hausdorff moment, depending on whether R⟶∞ or R<∞). Consequently, the normalization function Nz2 is either a generalized hypergeometric function (if R=∞) or a generalized hypergeometric polynomial of the degree M, if R<∞.Inserting the expression of the integration measure in the completion relation, to leave at the completion relation (or the identity operator decomposition) for Fock vectors, after angular integration, we obtain(19)∫0Rdz2hzpFqMai1p;bj1q;1/Cz2z2n=ρp,qba|n.The convergence radii for any kind of CSs are determined by calculating the limits (see, e.g., [20]):(20)R=1limn⟶∞ρp,qba|nn>0,(21)R=limn⟶∞ρp,qb/a|nρp,qb/a|n+1=1Climn⟶∞np−q−111+1/n=∞,ifp−q−1>0,1C<∞,ifp−q−1=0,0,ifp−q−1<0,and the CSs exist (that is, they make physical sense) only if R≠0.


## 3. Different Kinds of Coherent States

Generally, for a certain quantum system, there exist three kinds or manners to define the CSs: (a) the Barut–Girardello CSs (BG-CSs); (b) the Klauder–Perelomov CSs (KP-CSs); and (c) the Gazeau–Klauder CSs (GK-CSs). The corresponding expressions for these three kinds of CSs are different or divergent, except the case of HO-1D where these three definitions are convergent, i.e., their results are identical.To highlight, in turn, the most important characteristics of the three types of CSs, we will retain the following:(a)
The Barut–Girardello coherent states (BG-CSs) are defined as the eigenfunctions of the lowering operatorE^− (see [8]):(22)E^−zBG=zzBG,and their expansion in the Fock vector basis is(23)zBG=1NBGz2∑n=0∞znρp,qb/a|nn.The normalization functionNBGz2 is the following generalized hypergeometric function:(24)NBGz2=∑n=0∞z2nρp,qb/a|n=pFqai1p;bj1q;1Cz2.To find the weight functionhBGz of the integration measure, we appeal to the completion relations for the CSs, as well as for the Fock vectors. Then, we have (25)∑n,n′=0∞nn′ρp,qb/a|nρp,qb/a|n′∫0Rdz2z2nhBGzNBGz2∫02πdϕ2πz∗n′zn=1.The angular integral isz2nδnn′ and if we put h˜BGz≡hBGz/NBGz2, we have to solve an integral moment problem (Stieltjes or Hausdorff, depending on whether R=∞ or R<∞) [9]:(26)∫0Rdz2z2nh˜BGz=ρp,qba|n=Γp,qabCnΓn+1∑j=1qΓbj+n∑i=1pΓai+n.By changing the exponentn=s−1, we get(27)∫0Rdz2z2s−1h˜BGz=1CΓp,qabCsΓs∑j=1qΓbj−1+s∑i=1pΓai−1+s.Using the general relation for the classical integral to one Meijer’s G-function [21],(28)∫0∞dxxs−1Gp,qm,nωx|ai1n;ain+1pbj1m;bjm+1q=1ωs∏j=1mΓbj+s∏i=1nΓ1−ai−s∏j=m+1qΓ1−bj−s∏i=n+1pΓai+s,and identifying the constants, we obtain(29)h˜BGz=1CΓp,qaddz2Gp,q+1q+1,01Cz2|/;ai−11p0,bj−11q;/,so that finally the integration measure is(30)dμBGz=1CΓp,qabdϕ2πdz2NBGz2Gp,q+1q+1,01Cz2|/;ai−11p0,bj−11q;/.Replacing the expression ofh˜BGz in (27), we obtain an important integral of the real variable |z|2 (see [11]):(31)∫0Rdz2z2s−1Gp,q+1q+1,01Cz2|/;ai−11p0,bj−11q;/=CΓp,qa/bρp,qba|s−1.Due to their importance for the rest of this paper, we will call this equation thefundamental Barut–Girardello integral (f-BG-int).If we use (7) and (12) as well as the DOOT rules [17], we can write the BG-CSs as(32)zBG=1NBGz2∑n=0∞zE^+nρp,qb/a|n0=1NBGz2pFqai1p;bj1q;1CzE^+0,and the resolution of the unity operator (15) becomes(33)∫d2zπGp,q+1q+1,01Cz2|/;ai−11p0,bj−11q;/#pFqai1p;bj1q;1CzE^+××pFqai1p;bj1q;1Cz∗E^−#=CΓp,qa/b#pFqai1p;bj1q;1CE^+E^−#.This integral will be useful to calculate some real integrals involving the hypergeometric functions.If we perform the angular integral, we obtain(34)∫02πdϕ2π#pFqai1p;bj1q;1CzE^+pFqai1p;bj1q;1Cz∗E^−#=∑n=0∞#E^+E^−n#ρp,qb/a|n21C2z2n=#2pF2q+1ai1p,ai1p;1,bj1q,bj1q;1C2z2E^+E^−#.Then, it remains for us to perform the integral in the real space, with variablez2. Since in the DOOT formalism the operators (under the sign of integration) are treated as numbers, we can also replace, using DOOT rules, operators E^+ and E^− with numbers A and B, so that in the end, we get(35)∫0∞dz2Gp,q+1q+1,01Cz2|/;ai−11p0,bj−11q;/2pF2q+1ai1p,ai1p;1,bj1q,bj1q;ABC2z2=CΓp,qa/bpFqai1p;bj1q;ABC.(b)
The Klauder–Perelomov coherent states (KP-CSs) are defined as the result of the action of the generalized displacement unitary operator#expzE^+−z∗E^−# on the ground (or vacuum) state 0 (see [9]):(36)zKP=1NKPz2expzE^+0=1NKPz2∑n=0Mznn!E^+n0=1NKPz2∑n=0Mρp,qb/a|nn!znn.The normalization function is then(37)NKPz2=∑n=0Mρp,qb/a|nn!2zn=∑n=0M∏j=1qbjn∏i=1painCz2nn!=qFpMbj1q;ai1p;Cz2.Comparing with the normalization functionNBGz2of the BG-CSs, it is to observe the interchanging of the sets of parametersai1p and bj1q as well as the inversion of the constant in front of the real variable z2 of the generalized hypergeometric function.We must determine the expression of the weight functionhKPz of the integration measure dμKPz=d2z/πhKPz by the help of the resolution of the unity operator. Using the DOOT, this relation becomes(38)00∫0Rdz2hKPzNKPz2∫02πdϕ2π#expzE^+expz∗E^−#=1.After separate expansion of exponentials into power series, the angular integral becomes ([22])(39)∫02πdϕ2π#expzE^+expz∗E^−#=∑n=0∞#E^+E^−n#n!2z2n=#0F1;1;E^+E^−z2#=#I02zE^+E^−#.Following a successively similar path as in the case of BG-CSs, we will have(40)1#pFqMai1p;bj1q;1/CE^+E^−#∑n=0∞#E^+E^−n#n!2∫0Rdz2z2nh˜KPz=1.Hence, it follows that the integral must be of the moment problem type, withn=s−1:(41)∫0Rdz2z2s−1h˜KPz=Γs2ρp,qb/a|s−1=CΓp,qa/b1CsΓs∏i=1pΓai−1+s∏j=1qΓbj−1+s.Identifying the constants, after obtaining the solution of the above integral equation [21], the integration measure is finally written as(42)dμKPz=CΓp,qa/bdϕ2πdz2NKPz2Gq,p+1p+1,01Cz2|/;bj−11q0,ai−11p;/.Consequently, the decomposition relation of the unity operator turns into the following integral equation:(43)∫d2zπGq,p+1p+1,0Cz2|/;bj−11q0,ai−11p;/#expzE^+expz∗E^−#=1CΓp,qab#pFqai1p;bj1q;1CE^+E^−#.After performing the angular integral, we obtain the integral in the real space, with variablez2 (similarly, as for BG-CSs, we have used DOOT rules and replaced the operators E^+ and E^− with scalars A and B), so that in the end, we get(44)∫0∞dz2Gq,p+1p+1,0Cz2|/;bj−11q0,ai−11p;/I02zAB=Γp,qa/bCpFqai1p;bj1q;1CAB.Finally, let us point out that for these CSs, thefundamental Klauder–Perelomov integral (f-KP-int) is then(45)∫0Rdz2πz2s−1Gq,p+1p+1,0Cz2|/;bj−11q0,ai−11p;/=Γp,qa/bCΓs2ρp,qb/a|s−1.(c)
The Gazeau–Klauder coherent states (GK-CSs) were introduced by Gazeau and Klauder [10] and have the following expression:(46)J,γ=1NGKJ∑n=0∞Jnρp,qb/a|ne−iγenn,where0≤J≤∞ is a real number labelling the GK-CSs and −∞≤γ≤+∞ is a real characteristic parameter. The normalization function NGKJ is obtained, as usual, from the normalization condition J|J=1:(47)NGKJ=∑n=0∞Jnρp,qb/a|n=pFqai1p;bj1q;1CJ.We can see that they have the same mathematical structure asNBGz2.The integration measures(48)dμGKJ,γ=dγ2RdJhGKJmust be understood to satisfy the limit [10](49)∫dμGKJ,γ…=limR⟶∞∫−R+Rdγ2R…∫0∞dJhGKJ….The GK-CSs can be obtained if three steps are taken [19]:(i)
Firstly, let us define BG-CSs, but for thereal variable J, denoted by J:(50)E^−J=JJ.(ii)
Secondly, let us developJ into the Fock vector base, using the standard procedure:(51)J=1NGKJ∑n=0∞Jnρp,qb/a|nn.(iii)
The third step is to act with the exponential operatorexp−iγH^ on the state J (the parameter γ and the Hamiltonian H^ are considered less dimensional):(52)J,γ=exp−iγH^J=1NGKJ∑n=0MJnρp,qb/a|ne−iγenn.The resolution of the unity operator:(53)∫dμGKJ,γJ,γJ,γ=1.After some similar calculations as before, we arrive at the expression(54)∑n,n′=0∞nn′ρp,qb/a|nρp,qb/a|n′limR⟶∞∫0RdJhGKJNGKJJnJn′∫−R+Rdγ2Reiγen−en′=1.The integral with respect to the variableγ is δn,n′ and so we must solve the following moment problem:(55)∫0∞dJh˜GKJJn=ρp,qba|n,which is identical to that of BG-CSs (26). Consequently, the integration measure is(56)dμKPJ,γ=1CΓp,qabdγ2RdJNGKJGp,q+1q+1,01CJ/;ai−11p0,bj−11m;/.Let us replace this expression by the decomposition relation of the unity operator:(57)1CΓp,qab∑n,n′=0Mn><n′ρp,qb/a|nρp,qb/a|n′×limR⟶∞∫0RdJGp,q+1q+1,01CJ/;ai−11p0,bj−11m;/Jn+n′limR⟶∞12R∫−R+Rdγexp−iγen−en′=1.Performing the integral with respect toγ and using (17), finally we obtain(58)∫0∞dJJs−1Gp,q+1q+1,01CJ/;ai−11p0,bj−11q;/=CΓp,qa/bρp,qba|s−1,which is identical, from the mathematical point of view, to (31).An interesting situation arises in the case ofquantum systems with a linear energy spectrum, for which el=Cl, so p=q, ai1p=bj1q, and ρ0,0/|n=Cnn!.The GK-CSs for these systems are(59)J,γ=1NGKJ∑n=0MJnρp,qb/a|ne−iγCnn=1NGKJ∑n=0ME^+e−iγCJnρ0,0/|n0=1NGKJ0F0;;1CE^+e−iγCJ0=1NGKJexp1CE^+e−iγCJ0,where the normalization function is(60)NGKJ=exp1CJ00=1#0F0;;1/CE^+E^−#=#e−1CE^+E^−#.Let us check if the integral resulting from the decomposition of the unit operator is true:(61)limR⟶∞∫0RdJG0,11,01CJ|0;/∫−R+Rdγ2R#exp1CE^+e−iγCJ00exp1CE^−e+iγCJ#=1.Assuming that [23](62)G0,11,01CJ|0;/=exp1CJand developing in series the complex exponentials, after integration with respect to γ, it is easy to show that the above integral is true.By the way, linear energy spectrum systems also satisfy the conditions of temporal stability and action identity [10].


## 4. Results and Discussion

Let us turn our attention to the integrals which are called thefundamental CS integrals.i.e., equations (31), (45), and (58) are identical from a mathematical point of view. What differ are only the indices of Meijer’s G functions. Because f-BG-int and f-GK-int are the same, consequently, we will pay attention only to f-BG-int and f-KP-int. We will use these integrals to obtain some new generalized integrals involving Meijer’s and hypergeometric functions.First, let us consider a new hypergeometric function(63)rFsci1r;dj1s;Az2=∑m=0∞1ρr,sd/c|mAz2m,with the structure constants(64)ρr,sdc|m=1Amm!∏j=1sdjm∏i=1rcim.Let us calculate the integral(65)IGBG∗F≡∫0∞dz2Gp,q+1q+1,01Cz2/;ai−11p0,bj−11q;/rFsci1r;dj1s;Az2.If we write the functionrFsci1r;dj1s;Az2 as a power series, we obtain successively(66)IGBG∗F=∑m=0∞Amρr,sd/c|m∫0∞dz2z2mGp,q+1q+1,01Cz2/;ai−11p0,bj−11q;/=CΓp,qa/b∑m=0∞ρp,qb/a|mρr,sd/c|mAm=CΓp,qa/b∑m=0∞∏j=1qbjm∏i=1rcim1m∏i=1paim∏j=1sdjmCAmm!.Finally, we obtain a new integral, which is one of the goals of the present article:(67)∫0∞dz2Gp,q+1q+1,01Cz2/;ai−11p0,bj−11q;/rFsci1r;dj1s;Az2=CΓp,qa/bq+r+1Fp+sbj1q,ci1r,1;ai1p,dj1s;CA.To verify the correctness of this result, firstly we express the hypergeometric function through Meijer’s function (see [21]):(68)rFsci1r;dj1s;Az2=1Γr,sc/dGr,s+11,r−Az2|1−ci1r;/0;1−dj1s,and then we use classical Meijer’s integral for two G functions [15]:(69)∫0∞dz2z2α−1Gp,qm,n1Cz2|ai1n;ain+1pbj1m;bjm+1qGu,vs,t−Az2|ci1t;cit+1udj1s;djs+1v=1−AαGp+v,q+um+t,n+s−1CAai1n,bj1m,1−α−dj1s;1−α−djs+1v,ain+1p1−α−ci1t;1−α−cit+1u,bjm+1q,as well as the following transformations and argument simplifications of Meijer’s functions:(70)Gp,qm,n1z2|ai1pbj1q=Gq,pn,mz21−bj1q1−ai1p,(71)Gp,qm,nz2|α+ai1pα+bj1q=z2αGp,qm,nz2ai1pbj1q.Let us we calculate the same kind of integrals, but using f-KP-int, i.e.,(72)IGKP∗F≡∫0∞dz2Gq,p+1p+1,01Cz2/;bj−11q0,ai−11p;/rFsci1r;dj1s;Az2.Following the same procedure and steps after straightforward calculations, we obtain(73)∫0∞dz2Gq,p+1p+1,01Cz2/;bj−11q0,ai−11p;/rFsci1r;dj1s;Az2=Γp,qa/bCp+r+1Fq+sai1p,ci1r,1;bj1q,dj1s;CA.To these new kinds of integrals, we can also add the integrals previously obtained, which can be considered as a particular case of the previous:(74)∫0∞dz2Gp,q+1q+1,01Cz2/;ai−11p0,bj−11q;/2pF2q+1ai1p,ai1p;1,bj1q,bj1q;ABC2z2=CΓp,qa/bpFqai1p;bj1q;ABC.Let us examine some examples, to confirm the correctness of the obtained integrals.Example 1.
Let us consider the coefficientsp=1 and q=1, as well as a1=1, so we have(75)G1,22,01Cz2/;a1−10,b1−1;/=G0,11,01Cz2b1−1=1Cb1−1z2b1−1e−1/Cz2.
Integral (67) is then(76)∫0∞dz2G1,22,01Cz2/;a1−10,b1−1;/rFsci1r;dj1s;Az2=CΓb1r+1Fsb1,ci1r;dj1s;CA.
This result is in accordance with the integral 7.525.5, pp. 814 of Gradshteyn and Ryshik’s book [24]:(77)∫0∞dxe−xxs−1pFqa1,a2,…,ap;b1,b2,…,bq;az2=Γsp+1Fqs,a1,a2,…,ap;b1,b2,…,bq;a.
On the other hand, this equation can be regarded as the Laplace transform of the generalized hypergeometric function [25].Example 2.
Let us choosep=1, q=1, m=2, n=0, C=1, and a1,b1≠0. Then,(78)G1,22,0z2/;a1−10,b1−1;/=e−|z|2Ua1−b1,2−b1;z2,where Ua1−b1,2−b1;z2 is the Tricomi confluent hypergeometric function.
According to equation (4), we have(79)∫0∞dz2G1,22,0z2/;a1−10,b1−1;/rFsci1r;dj1s;Az2=Γb1r+2Fs+1b1,ci1r,1;a1,dj1s;A,which can be verified using the integral calculus [26]:(80)∫0∞dz2z2α−1e−z2Ua,b,z2=Γ1−b+αΓαΓa−b+α+1,max0,Reb−1<Reα.
Generally, by particularizing all coefficientsp, q, r, and s, as well as the set of parameters ai1p, bj1q, ci1r, and dj1s, it is possible to obtain a lot of new integrals (some of them unknown yet).


## 5. Conclusions

In the present paper, we try to show that, apart from the role that the formalism of coherent states plays in different branches of physics (quantum mechanics, quantum optics, and quantum information theory), the formalism of coherent states can also be useful in mathematics, in the field of special functions. Namely, to solve the equation of the decomposition of the unity operator (which constitutes the fundamental property of coherent states), we arrive at a new type of integrals. Beginning from these integrals, we have deduced a set of new integrals involving Meijer’s and generalized hypergeometric functions. After the particularization of their coefficientsp, q, r, and s, as well as the set of parameters ai1p, bj1q, ci1r, and dj1s, it is possible, on the one hand, to recover a lot of known integrals and, on the other hand, to obtain some new unknown integrals. The above calculation was possible because we used the diagonal operator ordering technique (DOOT). From the attached examples, it is to observe that, from the integration point of view, the creation E^+ and the annihilation E^− operators that appear under the integral’s sign can be treated as numbers, and consequently they can be replaced by simple numerical constants. This is another consequence of using the DOOT formalism.



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*Source: 1284378-2022-10-19.xml*