Inversion of a Class of Singular Integral Operators on Entire Functions
IINVERSION OF A CLASS OF SINGULAR INTEGRALOPERATORS ON ENTIRE FUNCTIONS
R. NASRI(*), A. SIMONIAN(*) AND F. GUILLEMIN (**)
Abstract.
Given constants x, ν ∈ C and the space H of entire functions in C vanishing at 0, we consider the integro-differential operator L = (cid:18) x ν (1 − ν )1 − x (cid:19) δ ◦ M , with δ = z d / d z and M : H → H defined by M f ( z ) = (cid:90) e − zt − ν (1 − (1 − x ) t ) f (cid:0) z t − ν (1 − t ) (cid:1) d tt , z ∈ C , for any f ∈ H . Operator L originates from an inversion problem in QueuingTheory. Bringing the inversion of L back to that of M translates into a singularVolterra integral equation, but with no explicit kernel.In this paper, the inverse of operator L is derived through a new inversionformula recently obtained for infinite matrices with entries involving Hyperge-ometric polynomials. For x / ∈ R − ∪ { } and Re( ν ) <
0, we then show that theinverse L − of L on H has the integral representation L − g ( z ) = 1 − x iπx e z (cid:90) (0+)1 e − xtz t ( t − g (cid:0) z ( − t ) ν (1 − t ) − ν (cid:1) d t, z ∈ C , for any g ∈ H , where the bounded integration contour in the complex planestarts at point 1 and encircles the point 0 in the positive sense. Other relatedintegral representations of L − are also provided. Introduction
The inversion of an integro-differential operator acting on entire functions in C isrelated to a new class of linear inversion formulas with coefficients involving Hyper-geometric polynomials. After an overview of the state-of-the-art in the associatedfields, we summarize our main contributions.1.1. Motivation.
Consider the following problem: let constants x ∈ ]0 , , ν < and the function R defined by (1.1) R ( ζ ) = x (1 − ζ ) − ν (1 − (1 − x ) ζ ) ν − , ζ ∈ [0 , . Let H be the linear space of entire functions in C vanishing at z = 0 and define the integro-differential operator L : H → H by (1.2) L f ( z ) = (cid:90) (cid:20) (1 + z R ( ζ )) f ( ζ R ( ζ ) · z ) − c z R ( ζ ) f (cid:48) ( ζ R ( ζ ) · z ) (cid:21) e − R ( ζ ) · z d ζ Date : Version of January 5, 2021. a r X i v : . [ m a t h . C A ] J a n R. NASRI(*), A. SIMONIAN(*) AND F. GUILLEMIN (**) for all z ∈ C , where f (cid:48) denotes the derivative of f ∈ H and with theconstant c in the integrand equal to c = 1 − νx − x . Given K ∈ H , solve the equation (1.3) L E ∗ ( z ) = K ( z ) , z ∈ C , for the unknown E ∗ ∈ H . This inversion problem has been motivated by an integral equation arising froma problem of Queuing Theory [1], namely, the study of the sojourn time in aProcessor-Sharing queue with batch customer arrivals.The operator L = L x,ν depends on parameters x and ν . Solving equation (1.3)for such parameters is thus equivalent to prove that this operator from H to itselfis onto. As detailed in this paper, the following Properties (I) and (II) for L andthe associated equation (1.3) can be successively outlined: (I) Reduction to a Linear System: power series expansions (1.4) E ∗ ( z ) = + ∞ (cid:88) (cid:96) =1 E (cid:96) z (cid:96) (cid:96) ! , K ( z ) = + ∞ (cid:88) b =1 ( − b K b z b b ! , z ∈ C , for a solution E ∗ ∈ H and the given K ∈ H reduce the resolution of(1.3) to that of the infinite lower-triangular linear system (1.5) ∀ b ∈ N ∗ , b (cid:88) (cid:96) =1 ( − (cid:96) (cid:18) b(cid:96) (cid:19) Q b,(cid:96) E (cid:96) = K b , with unknown E (cid:96) , (cid:96) ∈ N ∗ , and where the coefficient matrix Q = ( Q b,(cid:96) ) b,(cid:96) ∈ N ∗ ,on account of the specific function R introduced in (1.1), is given by (1.6) Q b,(cid:96) = Γ( b )Γ(1 − bν )Γ( b − bν ) xx − F ( (cid:96) − b, − bν ; − b ; x ) , (cid:54) (cid:96) (cid:54) b. In (1.6), Γ is the Euler Gamma function and F ( α, β ; γ ; · ) denotes the GaussHypergeometric function with complex parameters α , β , γ / ∈ − N . Recall that F ( α, β ; γ ; · ) reduces to a polynomial with degree − α (resp. − β ) if α (resp. β )equals a non positive integer; expression (1.6) for coefficient Q b,(cid:96) thus involves aHypergeometric polynomial with degree b − (cid:96) in both arguments x and ν .The diagonal coefficients Q b,b , b (cid:62)
1, are non-zero so that lower-triangular system(1.5) has a unique solution; equivalently, this proves the uniqueness of the solution E ∗ ∈ H to (1.3). To make this solution explicit in terms of parameters, writesystem (1.5) equivalently as(1.7) ∀ b ∈ N ∗ , b (cid:88) (cid:96) =1 A b,(cid:96) ( x, ν ) E (cid:96) = (cid:101) K b , with the reduced right-hand side ( (cid:101) K b ) defined by (cid:101) K b = Γ( b − bν )Γ( b )Γ(1 − bν ) x − x · K b , b (cid:62) , and with matrix A ( x, ν ) = ( A b,(cid:96) ( x, ν )) given by(1.8) A b,(cid:96) ( x, ν ) = ( − (cid:96) (cid:18) b(cid:96) (cid:19) F ( (cid:96) − b, − bν ; − b ; x ) , (cid:54) (cid:96) (cid:54) b. As recently shown [2], the linear relation (1.7) to which initial system (1.5) hasbeen recast can be explicitly inverted for any right-hand side ( K b ) b ∈ N ∗ , the inversematrix B ( x, ν ) = A ( x, ν ) − involving also Hypergeometric polynomials as well.This consequently solves system (1.5) explicitly, hence integral equation (1.3); (II) Factorization: operator L can be factored as (1.9) L = xν (1 − ν )1 − x · δ ◦ M where δ = z d / d z and M is the integral operator defined by (1.10) M f ( z ) = (cid:90) e − zt − ν (1 − (1 − x ) t ) f (cid:0) z t − ν (1 − t ) (cid:1) d tt , z ∈ C , for all f ∈ H . Using factorization (1.9), the resolution of (1.3) is thus equivalentto solving equation(1.11) M E ∗ = K with right-hand side K ( z ) = 1 − xν (1 − ν ) x · (cid:90) z K ( ζ ) ζ d ζ, z ∈ C , where K ∈ H as soon as K ∈ H . Integral equation (1.11) can be in turn recastinto the Volterra equation(1.12) (cid:90) (cid:98) τz Ψ (cid:18) z, ξz (cid:19) E ∗ ( ξ ) d ξ = z · K ( z ) , z ∈ C , for some constant (cid:98) τ and a kernel Ψ( z, · ). As Ψ( z, · ) has an integrable singularityof order Ψ( z, τ ) = O ( (cid:98) τ − τ ) − / near point τ = (cid:98) τ , (1.12) is therefore a singularVolterra integral equation of the first kind.1.2. State-of-the-art.
As equation (1.3) or (1.11) can be recast into the singularVolterra equation of the first kind (1.12), we here briefly review known results forthis class of integral equations.Given the constant α ∈ ]0 , (cid:90) z E ( ξ )( z − ξ ) α d ξ = κ ( z ) , z ∈ [0 , r ] , on a real interval [0 , r ], for the unknown function E and some given function κ (see[3, Chap.7], [4, Chap.2], [5, Chap.1]. If κ is absolutely continuous on [0 , r ], thenAbel’s equation has the unique solution E ∈ L [0 , r ] given by E ( z ) = sin( πα ) π · dd z (cid:20)(cid:90) z κ ( ξ )( z − ξ ) − α d ξ (cid:21) = sin( πα ) π (cid:20) κ (0) z − α + (cid:90) z κ (cid:48) ( ξ )( z − ξ ) − α d ξ (cid:21) , z ∈ [0 , r ] . (1.13) R. NASRI(*), A. SIMONIAN(*) AND F. GUILLEMIN (**)
This solution extends to a complex variable z ∈ C pertaining to a neighborhood ofpoint 0 where function κ is assumed to be analyti; the solution E is then analytic ina neighborhood of z = 0 if condition κ (0) = 0 holds, that is, if and only if κ ∈ H .More generally, let a compact subset Ω ⊂ C and a singular operator J : E (cid:55)→ J E defined by J E ( z ) = (cid:90) z N ( z, ξ ) E ( ξ ) d ξ, z ∈ Ω , where the kernel N verifies | N ( z, ξ ) | (cid:54) M | z − ξ | α , z, ξ ∈ Ω , z (cid:54) = ξ, for some constant M > α ∈ ]0 , J is known to be continuous (andalso compact) on space C [Ω] [6, Theorem 2.29]. No general results are available,however, on the inverse of J on a subspace of C [Ω].This standard framework may nevertheless suggest the existence of an integralrepresentation of the kind (1.13) for the entire solution to either equation (1.3)(1.11) or (1.12). In this paper, we will show how such integral representations canbe obtained for the solutions of these singular equations.1.3. Paper contribution.
The main contributions of this paper can be summa-rized as follows: (A) we first prove the above mentioned
Reduction Property I (Section 3.1)whereby integral equation (1.3) is reduced to linear system (1.5) with coefficientsrelated to Hypergeometric polynomials; (B) we next justify the
Factorization Property II (Section 3.2) for theintegro-differential operator L . We further specify how equation (1.11) can berecast into a Volterra integral equation with singular kernel (Section 3.3); (C) the previous results finally enable us to derive an integral representation ofthe inverse L − of operator L in space H in the form of the contour integral L − g ( z ) = 1 − x iπx e z (cid:90) (0+)1 e − xtz t ( t − g (cid:0) z ( − t ) ν (1 − t ) − ν (cid:1) d t, z ∈ C , for any g ∈ H , where the finite contour in the complex plane starts at 1 andencircles 0 in the positive sense (Section 4). By using suitable variable change inthe latter, other related integral representations of the inverse L − are also provided.2. Preliminaries
Infinite matrices inversion.
We first recall the results established in [2] forthe inversion of some class of lower-triangular matrices. These results will be usedbelow for the inversion of operator L .While stated for general matrices A ( x, ν ; α, β, γ ) depending on x , ν and threeother complex parameters α , β and γ [2, Theorem 2.3], we here only use the in-version property particularized to the matrix A ( x, ν ) introduced in (1.8) and cor-responding to the sub-case α = β = γ = 0. In such a case, the inversion formulafor lower-triangular matrices involving Hypergeometric polynomials F ( m, · ; · ; x ), m ∈ − N , can be stated as follows. Theorem 2.1. ( [2, Sect. 2.2] ) Let x, ν ∈ C and define the lower-triangularmatrices A ( x, ν ) and B ( x, ν ) by (2.1) A n,k ( x, ν ) = ( − k (cid:18) nk (cid:19) F ( k − n, − nν ; − n ; x ) ,B n,k ( x, ν ) = ( − k (cid:18) nk (cid:19) F ( k − n, kν ; k ; x ) for (cid:54) k (cid:54) n . The inversion formula (2.2) T n = n (cid:88) k =1 A n,k ( x, ν ) S k ⇐⇒ S n = n (cid:88) k =1 B n,k ( x, ν ) T k , n ∈ N ∗ , holds for any pair of complex sequences ( S n ) n ∈ N ∗ and ( T n ) n ∈ N ∗ . As a direct consequence of Theorem 2.1, a remarkable functional identity canbe derived for the exponential generating functions of sequences related by theinversion formula.
Corollary 2.1. ( [2, Sect. 3.2] ) Given sequences S and T related by the inver-sion formulae S = B ( x, ν ) · T ⇔ T = A ( x, ν ) · S , the exponential generatingfunction G ∗ S of the sequence S can be expressed by (2.3) G ∗ S ( z ) = exp( z ) · (cid:88) k (cid:62) ( − k T k z k k ! Φ( kν ; k ; − x z ) , z ∈ C , where Φ( α ; β ; · ) denotes the Confluent Hypergeometric function with pa-rameters α , β / ∈ − N . Parameters range.
Operator L has been initially introduced for real param-eters x ∈ ]0 ,
1[ and ν <
0. Following the results recalled in Section 2.1 and statedfor arbitrary complex parameters, we hereafter extend the definition (1.2) of L tocomplex values, namely • x ∈ C \ ( R − ∪ { } ) (so that 1 / (1 − x ) is finite and does not belong to theintegration interval [0 , • and ν ∈ C such that Re( ν ) < L ( H ) ⊂ H where H is againthe linear space of entire functions in C vanishing at 0. Remark 2.1.
Operator L , well-defined for Re( ν ) < , may not exist for othervalues of ν . In fact, consider the function f ∈ H defined by f ( z ) = z e (1 − x ) z , z ∈ C . By definition (1.2), it is easily verified that, for Re( ν ) < , its image L f ∈ H is given by L f ( z ) = − z − x Φ (cid:18) − ν ; 1 − ν ; − z (cid:19) , z ∈ C , Φ( · ; · ; · ) denoting the Kummer Confluent Hypergeometric function. For Re( ν ) > ,however, its image is given by L f ( z ) = − − νν (1 − x ) z ν Γ (cid:18) − ν ; z (cid:19) , z (cid:54) = 0 R. NASRI(*), A. SIMONIAN(*) AND F. GUILLEMIN (**) (where Γ( · ; · ) is the incomplete Gamma function), so that L f / ∈ H in this case. Properties of operator L Reduction to a linear system.
We have claimed in 1.1. (I) that the integro-differential equation (1.3) reduces to the infinite system (1.5). We justify thisassertion by showing how the coefficients of system (1.5) can be expressed in termsof Hypergeometric polynomials.
Proposition 3.1.
The Reduction Property (I) holds, that is, equation(1.3) reduces to system (1.5) with matrix Q = ( Q b,(cid:96) ) (cid:54) (cid:96) (cid:54) b related to Hy-pergeometric polynomials as given in (1.6). Proof.
To derive system (1.5), we expand both sides of (1.3) into power series ofvariable z and identify like powers on each side. The series expansion (1.4) of E ∗ ( Z )in powers of Z first provides(3.1) (1 + Z ) E ∗ ( ζZ ) − c Z d E ∗ d z ( ζZ ) = (cid:88) b (cid:62) Λ b ( ζ ) Z b b !where we set Λ b ( ζ ) = ζ b E b + b ζ b − E b − − b cζ b − E b for all ζ and with the constant c = (1 − νx ) / (1 − x ); applying equality (3.1) to the argument Z = R ( ζ ) · z , theintegrand of L E ∗ ( z ) in (1.2) can then be expanded into a power series of z as(3.2) L E ∗ ( z ) = (cid:90) (cid:20)(cid:88) b (cid:62) Λ b ( ζ ) ζ b R ( ζ ) b z b b ! (cid:21) e − R ( ζ ) z d ζ. Now, expanding the exponential e − R ( ζ ) z of the integrand in (3.2) into a powerseries of z gives the expansion(3.3) L E ∗ ( z ) = (cid:88) b (cid:62) ( − b z b b ! b (cid:88) (cid:96) =0 ( − (cid:96) (cid:18) b(cid:96) (cid:19) (cid:90) ζ (cid:96) Λ (cid:96) ( ζ ) R ( ζ ) b d ζ (after noting that Λ ( ζ ) = 0 since E = 0 by definition). On account of expansion(3.3) with the above definition (3.1) of Λ (cid:96) ( ζ ), together with the expansion (1.4) for K ( z ), the identification of like powers of these expansions readily yields the relation(3.4) b (cid:88) (cid:96) =1 ( − (cid:96) (cid:18) b(cid:96) (cid:19) B b,(cid:96) E ∗ (cid:96) + b (cid:88) (cid:96) =1 ( − (cid:96) (cid:18) b(cid:96) (cid:19) (cid:96) M b,(cid:96) − E (cid:96) − = K b , b (cid:62) , with B b,(cid:96) = M b,(cid:96) − (cid:96) c M b,(cid:96) − , where M b,(cid:96) denotes the definite integral(3.5) M b,(cid:96) = (cid:90) ζ (cid:96) R ( ζ ) b d ζ, (cid:54) (cid:96) (cid:54) b. By first changing the index in the second sum in the left-hand side of (3.4) and thenusing identity (cid:0) b(cid:96) +1 (cid:1) = ( b − (cid:96) ) · (cid:0) b(cid:96) (cid:1) / ( (cid:96) + 1), (3.4) reduces to (1.5) with coefficients(3.6) Q b,(cid:96) = ( (cid:96) + 1 − b ) M b,(cid:96) − (cid:96) c M b,(cid:96) − , (cid:54) (cid:96) (cid:54) b. The calculation of integral M b,(cid:96) in (3.5) in terms of Hypergeometric functions andits reduction to Hypergeometric polynomials is detailed in Appendix 5.1; this even-tually provides expression (1.6) for the coefficients of matrix Q = ( Q b,(cid:96) ). (cid:3) We can now deduce the unique solution to system (1.5).
Corollary 3.1.
Let ν ∈ C with Re( ν ) < . Given the sequence ( K b ) b (cid:62) , theunique solution ( E b ) b (cid:62) to system (1.5) is given by (3.7) E b = x − x b (cid:88) (cid:96) =1 ( − (cid:96) (cid:18) b(cid:96) (cid:19) F ( (cid:96) − b, (cid:96)ν ; (cid:96) ; x ) Γ( (cid:96) − (cid:96)ν )Γ( (cid:96) )Γ(1 − (cid:96)ν ) K (cid:96) for all b (cid:62) . Proof.
By expression (1.6) for the coefficients of lower-triangular matrix Q , equation(1.5) equivalently reads(3.8) b (cid:88) (cid:96) =1 ( − (cid:96) (cid:18) b(cid:96) (cid:19) F ( (cid:96) − b, − bν ; − b ; x ) · E (cid:96) = (cid:101) K b , (cid:54) (cid:96) (cid:54) b, when setting(3.9) (cid:101) K b = Γ( b − bν )Γ( b )Γ(1 − bν ) x − x · K b , b (cid:62) . The application of inversion Theorem 2.1 to lower-triangular system (3.8) readilyprovides the solution sequence ( E (cid:96) ) (cid:96) ∈ N in terms of the sequence ( (cid:101) K b ) b ∈ N ∗ ; using thentransformation (3.9), the final solution (3.7) for the sequence ( E (cid:96) ) (cid:96) ∈ N ∗ follows. (cid:3) Factorization of L . We now prove the factorization property 1.1. (II) forintegro-differential operator L . Proposition 3.2.
The Factorization Property (II) holds, that is, the linearoperator L on space H can be factored as in (1.9) in terms of operators δ = z d / d z and M . Proof.
Calculating the exponential generating function of the sequence ( − b K b , b (cid:62)
1, from relation (1.5) with help of (1.6) for the coefficients of matrix Q gives L E ∗ ( z ) = K ( z ) = (cid:88) b (cid:62) ( − b K b z b = (cid:88) b (cid:62) − Γ( b )Γ(1 − bν )Γ( b − bν ) x − x ( − z ) b b ! b (cid:88) (cid:96) =1 ( − (cid:96) (cid:18) b(cid:96) (cid:19) F ( (cid:96) − b, − bν ; − b ; x ) E (cid:96) , for all z ∈ C , that is, L E ∗ ( z ) = (cid:88) (cid:96) (cid:62) ( − (cid:96) (cid:96) ! E (cid:96) × (cid:88) b (cid:62) (cid:96) − Γ( b )Γ(1 − bν )Γ( b − bν ) x − x ( − z ) b ( b − (cid:96) )! F ( (cid:96) − b, − bν ; − b ; x )(3.10)(after changing the summation order on indexes b and (cid:96) ). Applying the generalidentity (5.14) to parameters m = b − (cid:96) (cid:62) β = − bν and γ = (cid:96) − bν + 1 to express R. NASRI(*), A. SIMONIAN(*) AND F. GUILLEMIN (**) polynomial F ( (cid:96) − b, − bν, − b ; x ) in terms of polynomial F ( (cid:96) − b, − bν, (cid:96) − bν +1; 1 − x ),we further obtain − Γ( b )Γ(1 − bν )Γ( b − bν ) F ( (cid:96) − b, − bν, − b ; x ) = − (1 − ν )Γ( (cid:96) + 1)Γ(1 − bν )Γ( (cid:96) − bν + 1) F ( (cid:96) − b, − bν, (cid:96) − bν + 1; 1 − x );(3.11)using the integral representation recalled in Appendix 5.1 - Equ.(5.1) for the factor F ( (cid:96) − b, − bν, (cid:96) − bν + 1; 1 − x ) in the right-hand side of (3.11) eventually yields − Γ( b )Γ(1 − bν )Γ( b − bν ) F ( (cid:96) − b, − bν, − b ; x ) = bν (1 − ν ) (cid:90) t − bν − (1 − t ) (cid:96) (1 − (1 − x ) t ) b − (cid:96) d t. Now, replacing the latter into the right-hand side of (3.10) provides(3.12) L E ∗ ( z ) = xν (1 − ν )1 − x (cid:88) (cid:96) (cid:62) ( − (cid:96) (cid:96) ! E (cid:96) · S (cid:96) ( z )where S (cid:96) ( z ) = (cid:88) b (cid:62) (cid:96) b ( b − (cid:96) )! · ( − z ) b (cid:90) t − bν − (1 − t ) (cid:96) (1 − (1 − x ) t ) b − (cid:96) d t. With the index change b (cid:48) = b − (cid:96) , the sum S (cid:96) ( z ) for given (cid:96) equivalently reads S (cid:96) ( z ) = (cid:88) b (cid:48) (cid:62) b (cid:48) + (cid:96)b (cid:48) ! · ( − z ) b (cid:48) + (cid:96) (cid:90) t − ( b (cid:48) + (cid:96) ) ν (1 − t ) (cid:96) (1 − (1 − x ) t ) b (cid:48) d tt = z dd z (cid:88) b (cid:48) (cid:62) b (cid:48) ! · ( − z ) b (cid:48) + (cid:96) (cid:90) t − ( b (cid:48) + (cid:96) ) ν (1 − t ) (cid:96) (1 − (1 − x ) t ) b (cid:48) d tt hence(3.13) S (cid:96) ( z ) = z dd z (cid:20) ( − z ) (cid:96) (cid:90) t − (cid:96)ν (1 − t ) (cid:96) d tt × exp (cid:0) − z t − ν [1 − (1 − x ) t ] (cid:1)(cid:21) . Replacing expression (3.13) into the left-hand side of (3.12), the linearity of operator δ = z d / d z and the permutation of the summation on index (cid:96) with the integrationwith respect to variable t ∈ [0 ,
1] enable us to obtain L E ∗ ( z ) = xν (1 − ν )1 − x · δ (cid:34)(cid:90) d tt e − z t − ν [1 − (1 − x ) t ] (cid:88) (cid:96) (cid:62) E (cid:96) (cid:96) ! z (cid:96) t − (cid:96)ν (1 − t ) (cid:96) (cid:35) , that is, L E ∗ ( z ) = xν (1 − ν )1 − x · δ (cid:34)(cid:90) d tt e − z t − ν [1 − (1 − x ) t ] × E ∗ (cid:0) z t − ν (1 − t ) (cid:1)(cid:35) = xν (1 − ν )1 − x · ( δ ◦ M ) E ∗ ( z ) , z ∈ C , for any function E ∗ ∈ H , as claimed in (1.9), with the corresponding definition ofintegral operator M on space H . (cid:3) The Volterra equation.
As outlined in the Introduction, the factorization(1.9) of operator L allows one to write equation (1.3) equivalently as(3.14) M E ∗ = K where K ∈ H relates to the initial function K as in (1.11). As the real function τ : t ∈ [0 , (cid:55)→ t − ν (1 − t ) has a unique maximum at point (cid:98) t = ν/ ( ν −
1) for ν <
0, we can introduce the variable changes t ∈ [0 , (cid:98) t ] (cid:55)→ τ − ( t ) = t − ν (1 − t ) and t ∈ [ (cid:98) t, (cid:55)→ τ + ( t ) = t − ν (1 − t ) on segments [0 , (cid:98) t ] and [ (cid:98) t, θ − = τ − − , θ + = τ − the respective inverse mappings of τ − and τ + , both defined on segment [0 , (cid:98) τ ] where (cid:98) τ = τ ( (cid:98) t ) = ( (cid:98) t ) − ν (1 − (cid:98) t ) (see illustration on Fig.1). The variable changes τ − and τ + then allow us to write equation (3.14) as a singular Volterra equation. 𝑡 −𝜈 (1 − 𝑡) 𝜏 𝜃 − (𝜏) 𝜃 + (𝜏) 𝜏̂ 𝑡 𝑡̂ Figure 1.
Graph of function τ : t ∈ [0 , (cid:55)→ t − ν (1 − t ) (here ν = − for illustration). Corollary 3.2.
Given constants x and ν as above, the equivalent equation(3.14) can be recast into the singular Volterra integral equation (3.16) (cid:90) (cid:98) τ z (cid:20) Ψ − (cid:18) z, ξz (cid:19) − Ψ + (cid:18) z, ξz (cid:19)(cid:21) E ∗ ( ξ ) d ξ = z · K ( z ) , z ∈ C , where we set Ψ ± ( z, τ ) = e − zθ ± ( τ ) − ν (1 − (1 − x ) θ ± ( τ )) θ ± ( τ ) − ν ( − ν + ( ν − θ ± ( τ )) , (cid:54) τ (cid:54) (cid:98) τ , with θ ± introduced in (3.15), and where K ∈ H is defined by (1.11). We refer to Appendix 5.2 for the proof of Corollary 3.2. It is noted there thatthe kernel τ (cid:55)→ Ψ − ( z, τ ) − Ψ + ( z, τ ) of Volterra equation (3.16) is singular with anintegrable singularity at the boundary τ = (cid:98) τ of order O ( (cid:98) τ − τ ) − / . Although giving a reformulation to initial equation (1.3), equation (3.16) remainsdifficult to solve as its kernel depends on inverse functions θ − and θ + which cannotbe made explicit simply. 4. Inversion of operator L We now provide integral representations for the inverse of operator L on H or, equivalently, integral representations for the solution of integral equation (1.3)addressed in the Introduction. Theorem 4.1.
Let ν ∈ C with Re( ν ) < . Then a) the operator L : H → H is a bijection; b) given K ∈ H , the unique solution E ∗ = L − K ∈ H to the integralequation (1.3) has the integral representation E ∗ ( z ) = L − K ( z )= 1 − x iπx e z (cid:90) (0+)1 e − xtz t ( t − K (cid:0) z ( − t ) ν (1 − t ) − ν (cid:1) d t, z ∈ C , (4.1) where the contour in integral (4.5) in variable t is a loop starting andending at point t = 1 , and encircling the origin t = 0 once in the positivesense (see Fig.2, red solid line). • • Plane ℂ 𝑡 Figure 2.
Integration contours around points 0 and 1.Proof. a) Given K ∈ H , equation (1.3) for E ∗ ∈ H is equivalent to system (1.5)for the coefficients ( E (cid:96) ) (cid:96) (cid:62) of the exponential series expansion of E ∗ . For Re( ν ) < L : H → H is consequently one-to-one and onto, and has aninverse L − on H . b) An integral representation for the inverse operator L − is now derived asfollows. Setting S = E and T = (cid:101) K in (2.3), with the sequence (cid:101) K = ( (cid:101) K b ) b (cid:62) defined as in (3.9), we obtain G ∗ E ( z ) = e z · (cid:88) b (cid:62) ( − b (cid:101) K b z b b ! Φ( bν ; b ; − x z )= x − x e z · (cid:88) b (cid:62) ( − b Γ( b − bν )Γ( b )Γ(1 − bν ) · K b z b b ! Φ( bν ; b ; − x z )(4.2)after using (3.9) to express (cid:101) K b in terms of K b , b (cid:62)
1. Invoke then the integralrepresentation(4.3) Φ( α ; β ; Z ) = − iπ Γ(1 − α )Γ( β )Γ( β − α ) (cid:90) (0+)1 e Zt ( − t ) α − (1 − t ) β − α − d t of the Confluent Hypergeometric function Φ( α ; β ; · ) for Re( β − α ) > α = bν and β = b ∈ N ∗ with Re( ν ) <
1, expression (4.2) nowreads G ∗ E ( z ) = 1 − x iπ x e z (cid:90) (0) + e − xzt d tt ( t − (cid:88) b (cid:62) ( − b K b b ! ( z ( − t ) ν (1 − t ) − ν ) b = 1 − x iπx e z (cid:90) (0+)1 e − xtz t ( t − K (cid:0) z ( − t ) ν (1 − t ) − ν (cid:1) d t (4.4)for all z ∈ C . As E ∗ ( z ) = G ∗ E ( z ) by definition, expression (4.4) readily yields thefinal representation (4.1), as claimed. (cid:3) As mentioned in the latter proof, the representation (4.1) of the inverse L − isactually valid for Re( ν ) <
1, although the operator L is defined on space H forRe( ν ) < L − with Re( ν ) < Corollary 4.1.
Let ν ∈ C with Re( ν ) < .Given K ∈ H , the unique solution E ∗ = L − K ∈ H to integral equa-tion (1.3) has the equivalent integral representations E ∗ ( z ) = L − K ( z )= 1 − x iπx e (1 − x ) z (cid:90) (1+)0 e xtz t (1 − t ) K (cid:0) z t − ν ( t − ν (cid:1) d t, z ∈ C , (4.5) where the contour in (4.5) in variable t is a loop starting and endingat point t = 0 , and encircling point t = 1 once in the positive sense (seeFig.2, blue dotted line), and E ∗ ( z ) = L − K ( z )= 1 − x iπ x e (1 − x ) z (cid:90) c + i ∞ c − i ∞ e x zr − r K (cid:18) z (1 − r ) ν r (cid:19) d r, z ∈ C , (4.6) where the contour in (4.6) is the vertical line Re( r ) = c , for any realabcissa < c < (see Fig.3, red dotted line). 𝑟 = 𝑡1 − 𝑡 + 1 𝑐 ℂ 𝑟 ℂ 𝑡 • • • • • • Figure 3.
Transformed integration contours.Proof. • By the variable change t (cid:55)→ − t , formula (4.1) readily entails (4.5) whichis defined for Re( ν ) < • Let 0 < c <
1. As a contour in integral (4.1), choose the circle centeredat point 1 − / (2 c ) on the real axis and with radius 1 / (2 c ); this circle passesthrough point 1 and encircles the origin (see Fig.3). It is easily verified that thehomographic transformation t (cid:55)→ r (cid:48) = t/ (1 − t ) maps this circle 1-to-1 and onto thevertical line Re( r (cid:48) ) = c −
1. Applying the latter variable change t (cid:55)→ r (cid:48) to (4.1)with ( − t ) ν (1 − t ) − ν = ( − r (cid:48) ) ν r (cid:48) , d t = d r (cid:48) (1 + r (cid:48) ) then readily gives L − K ( z ) = (cid:18) − x iπx (cid:19) e z (1 − x ) (cid:90) Re( r (cid:48) )= c − e z x r (cid:48) K (cid:18) z ( − r (cid:48) ) ν r (cid:48) (cid:19) d r (cid:48) r (cid:48) for all z ∈ C . The mapping r (cid:48) (cid:55)→ r = r (cid:48) + 1 then eventually transforms the latterintegral to the expected representation (4.6) with integration contour the verticalline Re( r ) = c , 0 < c < (cid:3) By the factorization (1.9), we readily deduce that the inverse of operator M isgiven by(4.7) M − g ( z ) = x ν (1 − ν )1 − x · L − ( z g (cid:48) )( z ) , z ∈ C , for all g ∈ H , with inverse L − provided by either integral representation (4.1),(4.5) or (4.6). The involvement of the derivative g (cid:48) for the inverse M − g in (4.7)reminds us of formula (1.13) in the particular case of the Abel’s equation. References [1] Guillemin F, Quintuna Rodriguez VK, Simonian A, Nasri R.
Sojourn time in a M [ X ] /M/ Processor Sharing Queue with Batch Arrivals (II) . arXiv preprint arXiv:2006.02198; 2020.[2] Nasri R, Simonian A, Guillemin F.
A New Linear Inversion Formula for a Classof Hypergeometric Polynomials , Integral Transforms and Special Functions , 2020,https://doi.org/10.1080/10652469.2020.1833002.[3] Bitsadze AV.
Integral Equations of First Kind , ed. World Scientific, 1995[4] Estrada R, Kanwal RP.
Singular Integral Equations , ed. Birh¨auser, 2000 [5] Gorenflo R and Vessella S. Abel Integral Equations, Analysis and Applications , Lecture Notesin Mathematics 1461, ed. Springer, 1991[6] Kress R.
Linear Integral Equations , Third edition, ed. Springer 2014[7] Erdelyi A.
Higher Transcendental Functions , Vol.1, ed. MacGraw Hill, 1981[8] Olver FW, Lozier DW, Boisvert RF, et al. (ed.).
NIST Handbook of Mathematical Functions .Cambridge university press. 2010.[9] Gradsteyn IS, Ryzhik IM.
Table of Integrals, Series and Products . ed. Academic Press. 2007. Appendix
Proof of Proposition 3.1 (continued).
We conclude the proof of Propo-sition 3.1 by expressing the coefficients Q b,(cid:96) , 1 (cid:54) (cid:96) (cid:54) b , introduced in (3.6) interms of Hypergeometric polynomials only. We first calculate coefficients Q b,(cid:96) ( s ),1 (cid:54) (cid:96) (cid:54) b , in terms of the general Gauss Hypergeometric function F . Recall that F = F ( α, β ; γ ; · ) has the integral representation [8, Chap.15, Sect.15.6.1](5.1) F ( α, β ; γ ; z ) = Γ( γ )Γ( β )Γ( γ − β ) (cid:90) t β − (1 − t ) γ − β − (1 − zt ) α d t, | z | < , for real parameters α , β , γ where γ > β > Lemma 5.1.
We have Q b,(cid:96) = − Γ( (cid:96) )Γ(1 − b ν )Γ( (cid:96) + 1 − b ν ) (cid:18) x b − x (cid:19) (cid:104) ν ( b − (cid:96) ) × F ( b (1 − ν ) , (cid:96) ; (cid:96) + 1 − b ν ; 1 − x ) + ( (cid:96) − b ν ) × F ( b (1 − ν ) , (cid:96) ; (cid:96) − b ν ; 1 − x ) (cid:105) (5.2) for (cid:54) (cid:96) (cid:54) b . Proof.
To calculate the integral M b,(cid:96) introduced in (3.5), use the definition of R ( t ),to write M b,(cid:96) = x b (cid:90) t (cid:96) (1 − t ) − b ν (1 − (1 − x ) t ) b ( ν − d t ;using representation (5.1) for parameters α = − b (1 − ν ), β = (cid:96) + 1, γ = 2 + (cid:96) − b ν ,this integral reduces to(5.3) M b,(cid:96) = Γ( (cid:96) + 1)Γ(1 − b ν )Γ(2 + (cid:96) − b ν ) x b F ( b (1 − ν ) , (cid:96) + 1; 2 + (cid:96) − b ν ; 1 − x );after (5.3) and the expression (3.6) of coefficient Q b,(cid:96) , we then derive Q b,(cid:96) = Γ( (cid:96) )Γ(1 − b ν )Γ( (cid:96) + 1 − b ν ) x b × (cid:104) (cid:96)(cid:96) + 1 − b ν ( (cid:96) + 1 − b ) · F ( b (1 − ν ) , (cid:96) + 1; (cid:96) + 2 − b ν ; 1 − x ) − (cid:96) c · F ( b (1 − ν ) , (cid:96) ; (cid:96) + 1 − b ν ; 1 − x ) (cid:105) . (5.4)To simplify further the latter expression, first invoke the identity(5.5) β F ( α, β + 1; γ + 1; z ) = γ F ( α, β ; γ ; z ) − ( γ − β ) F ( α, β ; γ + 1; z )easily derived from representation (5.1) for F ( α, β + 1; γ + 1; z ), after splittingthe factor t β of the integrand into t β = t β − − t β − (1 − t ). Applying (5.5) to α = b (1 − ν ), β = (cid:96) and γ = (cid:96) + 1 − b ν then enables one to express the term F ( b (1 − ν ) , (cid:96) + 1; (cid:96) + 2 − b ν ; 1 − x ) in the r.h.s. of (5.4) as a combination of F ( b (1 − ν ) , (cid:96) ; (cid:96) + 1 − b ν ; 1 − x ) and F ( b (1 − ν ) , (cid:96) ; (cid:96) + 2 − b ν ; 1 − x ) hence, aftersimple algebra, Q b,(cid:96) = Γ( (cid:96) )Γ(1 − b ν )Γ( (cid:96) + 1 − b ν ) x b (cid:104) { ( (cid:96) + 1 − b ) − (cid:96) c } · F ( b (1 − ν ) , (cid:96) ; (cid:96) + 1 − b ν ; 1 − x ) − ( (cid:96) + 1 − b )(1 − b ν ) (cid:96) + 1 − b ν · F ( b (1 − ν ) , (cid:96) ; (cid:96) + 2 − b ν ; 1 − x ) (cid:105) . (5.6)Furthermore, the contiguity identity [8, Sect.15.5.18] γ [ γ − − (2 γ − α − β − z ] F ( α, β ; γ ; z ) +( γ − α )( γ − β ) z F ( α, β ; γ + 1; z ) = γ ( γ − − z ) F ( α, β ; γ − z )(5.7)applied to α = b (1 − ν ), β = (cid:96) and γ = (cid:96) + 1 − b ν allows us to write the last term F ( b (1 − ν ) , (cid:96) ; (cid:96) + 2 − b ν ; 1 − x ) in the bracket of the r.h.s. of (5.6) as a combinationof F ( b (1 − ν ) , (cid:96) ; (cid:96) − b ν ; 1 − x ) and F ( b (1 − ν ) , (cid:96) ; (cid:96) + 1 − b ν ; 1 − x ), that is, F ( b (1 − ν ) , (cid:96) ; (cid:96) + 2 − b ν ; 1 − x ) = (cid:96) + 1 − b ν ( (cid:96) + 1 − b )(1 − b ν )(1 − x ) × (cid:104) ( (cid:96) − b ν ) x · F ( b (1 − ν ) , (cid:96) ; (cid:96) − b ν ; 1 − x ) − [ (cid:96) − b ν − ( (cid:96) + 1 − b − b ν )(1 − x )] · F ( b (1 − ν ) , (cid:96) ; (cid:96) + 1 − b ν ; 1 − x ) (cid:105) ;inserting the latter relation into the right-hand side of (5.6) then yields Q b,(cid:96) = Γ( (cid:96) )Γ(1 − b ν )Γ( (cid:96) + 1 − b ν ) x b × (cid:104) T b,(cid:96) · F ( b (1 − ν ) , (cid:96) ; (cid:96) + 1 − b ν ; 1 − x ) − − x ( (cid:96) − b ν ) x · F ( b (1 − ν ) , (cid:96) ; (cid:96) − b ν ; 1 − x ) (cid:105) (5.8)where T b,(cid:96) = b ν − (cid:96) c + ( (cid:96) − b ν )1 − x = ( (cid:96) − b ) νx − x after the definition c = (1 − νx ) / (1 − x ) of constant c . Inserting this value of T b,(cid:96) in the right-hand side of (5.8) readily provides expression (5.2) for Q b,(cid:96) . (cid:3) We finally show how coefficient Q b,(cid:96) can be written in terms of a Hypergeometricpolynomial only. Applying the general identity [9, Chap.9, Sect.9.131.1](5.9) F ( α, β ; γ ; z ) = (1 − z ) γ − α − β F ( γ − α, γ − β ; γ ; z ) , | z | < , to each term F ( b (1 − ν ) , (cid:96) ; (cid:96) + 1 − bν ; 1 − x ) and F ( b (1 − ν ) , (cid:96) ; (cid:96) − bν ; 1 − x ) in (5.2),we obtain(5.10) Q b,(cid:96) = − Γ( (cid:96) )Γ(1 − b ν )Γ( (cid:96) + 1 − b ν ) · x b (1 − x ) ( (cid:96) − bν ) × R b,(cid:96) , b (cid:62) (cid:96) (cid:62) , where we set R b,(cid:96) = bν ( b − (cid:96) ) (cid:96) − bν · F ( (cid:96) − b + 1 , − bν + 1; (cid:96) − bν + 1; 1 − x ) + b x − · F ( (cid:96) − b, − bν ; (cid:96) − bν ; 1 − x ) . From the identity [8, Chap.15, Sect.15.5.1](5.11) dd z F ( α, β ; γ ; z ) = αβγ F ( α + 1 , β + 1; γ + 1; z ) , | z | < , applied to parameters α = (cid:96) − b , β = − bν and γ = (cid:96) − bν , the factor R b,(cid:96) abovethen equals the derivative R b,(cid:96) = x b dd z (1 − z ) − b F ( (cid:96) − b, − bν ; (cid:96) − bν ; z ) | z =1 − x = x b dd z F ( b − bν, (cid:96) ; (cid:96) − bν ; z ) | z =1 − x = x b ( b − bν ) (cid:96)(cid:96) − bν F ( b − bν + 1 , (cid:96) + 1; (cid:96) − bν + 1; 1 − x )hence(5.12) R b,(cid:96) = ( b − bν ) (cid:96)(cid:96) − bν x − F ( (cid:96) − b, − bν ; (cid:96) − bν + 1; 1 − x )where we have successively applied identity (5.9), (5.11) and (5.9) again to derivethe second, third and fourth equality, respectively. Using (5.12), expression (5.10)for Q b,(cid:96) then reads(5.13) Q b,(cid:96) = − Γ( (cid:96) )Γ(1 − bν )Γ( (cid:96) − bν ) x − x (cid:96) (1 − ν ) (cid:96) − bν S b,(cid:96) ( ν ; 1 − x ) , b (cid:62) (cid:96) (cid:62) , where we set S b,(cid:96) ( ν ; 1 − x ) = F ( (cid:96) − b, − bν ; (cid:96) − bν + 1; 1 − x ) . To reduce further S b,(cid:96) ( ν ; 1 − x ), invoke the identity [8, Chap.15, Sect.15.8.7](5.14) F ( − m, β, γ ; 1 − x ) = Γ( γ )Γ( γ − β + m )Γ( γ − β )Γ( γ + m ) F ( − m, β, β +1 − m − γ ; x ) , x ∈ C , for any non negative integer m and complex numbers β , γ such that Re( γ ) > Re( β );applying (5.14) to factor S b,(cid:96) ( ν ; 1 − x ) in (5.13) then readily gives the final expression(1.6) for all indexes b (cid:62) (cid:96) (cid:62)
1. This concludes the proof of Proposition 3.1 (cid:4)
Proof of Corollary 3.2. • From the definition (1.10) of integral operator M ,split the integral M E ∗ ( z ) = (cid:90) e − z t − ν (1 − (1 − x ) t ) E ∗ (cid:0) zt − ν (1 − t ) (cid:1) d tt = (cid:90) (cid:98) t ( ... ) d tt + (cid:90) (cid:98) t ( ... ) d tt over adjacent segments [0 , (cid:98) t ] and [ (cid:98) t, τ = t − ν (1 − t ) on each of these two intervals with τ = τ − ( t ) ⇔ t = θ − ( τ ) ∈ [0 , (cid:98) t ]and τ = τ + ( t ) ⇔ t = θ + ( τ ) ∈ [ (cid:98) t,
1] by the definition (3.15) of mappings θ − and θ + ,we then successively obtain M E ∗ ( z ) = (cid:90) (cid:98) τ e − zθ − ( τ ) − ν (1 − (1 − x ) θ − ( τ )) E ∗ ( z τ ) − d τθ − ( τ ) − ν ( ν + (1 − ν ) θ − ( τ ))+ (cid:90) (cid:98) τ e − zθ + ( τ ) − ν (1 − (1 − x ) θ + ( τ )) E ∗ ( z τ ) − d τθ + ( τ ) − ν ( ν + (1 − ν ) θ + ( τ )) with (cid:98) τ = τ − ( (cid:98) t ) = τ + ( (cid:98) t ) and the differential d t/t = − d τ / [ t − ν ( ν + (1 − ν ) t )]; thisreadily reduces to a single integral over segment [0 , (cid:98) τ ], that is, M E ∗ ( z ) = (cid:90) (cid:98) τ [Ψ − ( z, τ ) − Ψ + ( z, τ )] E ∗ ( z τ ) d τ with Ψ − ( z, τ ) and Ψ + ( z, τ ) given as in the Corollary. The final variable change ξ = z · τ yields the right-hand side of (3.16) and the corresponding integral equation. • We finally verify that the r.h.s. of (3.16) is well-defined for any E ∗ ∈ H . Thedenominator t − ν ( − ν + ( ν − t ) of Ψ − ( z, τ ) with t = θ − ( τ ) (resp. of Ψ + ( z, τ ) with t = θ + ( τ )) vanishes at either τ = 0 or τ = (cid:98) τ (resp. at τ = (cid:98) τ ). As to the possiblesingularity at τ = 0 for Ψ − ( z, τ ), we have τ ∼ t − ν for small t = θ − ( τ ) so that1 t − ν ( − ν + ( ν − t ) ∼ − t ν ν ∼ − ντ , τ → E ∗ ( zτ ) · Ψ − ( z, τ ) is thus integrable near τ = 0 for any E ∗ ∈ H .Furthermore, a second order Taylor expansion of τ = τ ( t ) near point t = (cid:98) t yields τ = (cid:98) τ + τ (cid:48)(cid:48) ( (cid:98) t ) ( t − (cid:98) t ) / o ( t − (cid:98) t ) with τ (cid:48) ( (cid:98) t ) = 0 by definition and τ (cid:48)(cid:48) ( (cid:98) t ) <
0; as aresult, t − (cid:98) t ∼ ± (cid:115) − (cid:98) τ − τ ) τ (cid:48)(cid:48) ( (cid:98) t ) , τ → (cid:98) τ . The denominator t − ν ( − ν + ( ν − t ) of either Ψ − ( z, τ ) or Ψ + ( z, τ ) is consequentlyasymptotic to t − ν ( − ν + ( ν − t ) ∼ ( (cid:98) t ) − ν ( ν − t − (cid:98) t ) ∼ ± ( (cid:98) t ) − ν ( ν − (cid:115) (cid:98) τ − τ ) − τ (cid:48)(cid:48) ( (cid:98) t )when τ → (cid:98) τ ; the singularity of Ψ − ( z, τ ) (resp. Ψ + ( z, τ )) at point τ = (cid:98) τ is conse-quently of orderΨ − ( z, τ ) = O (cid:18) √ (cid:98) τ − τ (cid:19) , Ψ + ( z, τ ) = O (cid:18) √ (cid:98) τ − τ (cid:19) and the kernel Ψ( z, · ) = Ψ (cid:48)− ( z, · ) − Ψ + ( z, · ) is thus integrable at τ = (cid:98) τ . This ensuresthat the singular integral (3.16) is well-defined for any E ∗ ∈ H (cid:4) Address: (*) Orange Labs, OLN/NMP, Orange Gardens, 44 avenue de la R´epublique,CS 50010, 92326 Chatillon Cedex, France France (**) Orange Labs Networks Lannion,2 avenue Pierre Marzin, 22307 Lannion Cedex, Lannion, France
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