A method of composition orthogonality and new sequences of orthogonal polynomials and functions for non-classical weights
aa r X i v : . [ m a t h . C A ] J un A METHOD OF COMPOSITION ORTHOGONALITY AND NEW SEQUENCES OFORTHOGONAL POLYNOMIALS AND FUNCTIONS. THE SQUARE OF MACDONALDFUNCTION WEIGHT CASE
S. YAKUBOVICH
DEPARTMENT OF MATHEMATICS, FAC. SCIENCES OF UNIVERSITY OF PORTO,RUA DO CAMPO ALEGRE, 687; 4169-007 PORTO (PORTUGAL) A BSTRACT . A new method of composition orthogonality is introduced. It is applied to generate new sequencesof orthogonal polynomials and functions. In particular, classical orthogonal polynomials are interpreted in thesense of composition orthogonality. Finally, new sequences of orthogonal polynomials with respect to the weightfunction x α ρ ν ( x ) , ρ ν ( x ) = x ν / K ν ( √ x ) , x > , ν ≥ , α > −
1, where K ν ( z ) is the modified Bessel functionor Macdonald function, are investigated. Differential properties, recurrence relations, explicit representations,generating functions and Rodrigues-type formulae are obtained. The corresponding multiple orthogonal polyno-mials are exhibited.
1. I
NTRODUCTION AND PRELIMINARY RESULTS
In studying Prudnikov’s sequence of orthogonal polynomials (see [10]) with the weight function x α ρ ν ( x ) ,where ρ ν ( x ) = x ν / K ν ( √ x ) , x > , ν ≥ , α > − K ν ( z ) is the Macdonald or modified Bessel func-tion [1], Vol. II the author interpreted this sequence in terms of the Laguerre composition orthogonality,involving the differential operator θ = xDx , where D = d / dx . The main aim of this paper is to extend themethod, employing classical orthogonal polynomials (Hermite, Laguerre, Jacobi) [1] to find new sequencesof orthogonal polynomials with non-classical weights such as, for instance, the square of Macdonald func-tion and other hypergeometric functions. In fact, we will define the composition orthogonality in the follow-ing way. Definition 1.
Let ω ( t ) , ϕ ( t ) t ∈ [ a , b ] , − ∞ ≤ a < b ≤ ∞ be nonnegative functions, θ = tDt. Let f , gbe complex-valued functions such that the product of two operators f ( θ ) g ( θ ) commutes. Then f , g arecompositionally orthogonal with respect to the measure ω ( t ) dt relatively to the function ϕ if Z ba f ( θ ) g ( θ ) { ϕ ( t ) } ω ( t ) dt = . ( . ) We will see in the sequel that this definition suits well with the vector space of polynomials over R , whenthe composition orthonormality of the sequence { P n } n ≥ is defined by Date : June 23, 2020.2000
Mathematics Subject Classification.
Key words and phrases.
Classical orthogonal polynomials, Macdonald function, Tricomi function, generalized hypergeometricfunction.E-mail: [email protected] work was partially supported by CMUP, which is financed by national funds through FCT(Portugal), under the project withreference UIDB/00144/2020. Z ba P n ( θ ) P m ( θ ) { ϕ ( t ) } ω ( t ) dt = δ n , m , ( . ) where δ n , m , n , m ∈ N is the Kronecker symbol. Moreover, for some class of functions ϕ it is possible totransform the left-hand side of the equality (1.2) to the usual orthogonality with respect to a new weightfunction. Then since the n -th power of the operator θ satisfies the Viskov-type identity [8] θ n = ( xDx ) n = x n D n x n , n ∈ N , ( . ) we can get via integration by parts new properties of the sequence { P n } n ≥ and its relationship, for instance,with classical orthogonal polynomials. In fact, let us consider a class of functions ϕ representable by themodified Laplace transform of some nonnegative function ψϕ ( t ) = t Z ∞ e − x / t ψ ( x ) dx , t ∈ [ a , b ] ⊂ ( , ∞ ) . ( . ) Hence it is easily seen that θ k n t − e − x / t o = ( tDt ) k n t − e − x / t o = x k t − e − x / t , k ∈ N , ( . ) and therefore, differentiating under the integral sign in (1.4), we derive θ k { ϕ ( t ) } = t Z ∞ e − x / t x k ψ ( x ) dx , k ∈ N . ( . ) It is indeed allowed, for instance, due to the assumed convergence of the integral Z ∞ e − x / b x k ψ ( x ) dx < ∞ , k ∈ N . ( . ) Consequently, returning to (1.2) we write its left-hand side in the form Z ba P n ( θ ) P m ( θ ) { ϕ ( t ) } ω ( t ) dt = Z ba Z ∞ e − x / t P n ( x ) P m ( x ) ψ ( x ) ω ( t ) dxdtt . ( . ) The interchange of the order of integration on the right-hand side in (1.8) is permitted by Fubini’s theoremunder the imposed condition Z ba Z ∞ e − x / t x k ψ ( x ) ω ( t ) dxdtt < ∞ , k ∈ N , ( . ) and, combining with (1.2), we find the equalities Z ba P n ( θ ) P m ( θ ) { ϕ ( t ) } ω ( t ) dt = Z ∞ P n ( x ) P m ( x ) ψ ( x ) Ω ( x ) dx = δ n , m , ( . ) where Ω ( x ) = Z ba e − x / t ω ( t ) dtt , x > . ( . ) Thus we see that the sequence { P n } n ≥ is orthonormal over ( , ∞ ) with respect to the measure Ω ( x ) dx .Moreover, one can consider the left-hand side of the first equality in (1.10) as an inner product on the vectorspace of polynomials over R (a pre-Hilbert space) h p , q i = Z ba p ( θ ) q ( θ ) { ϕ ( t ) } ω ( t ) dt , ( . ) omposition orthogonality and new sequences of orthogonal polynomials and functions 3 inducing the norm by the equality || p || = p h p , p i = (cid:18) Z ∞ p ( x ) ψ ( x ) Ω ( x ) dx (cid:19) / . ( . )
2. T
HE USE OF CLASSICAL ORTHOGONAL POLYNOMIALS
Laguerre polynomials.
We begin to consider sequences of orthogonal polynomials which are compo-sitionally orthogonal with respect to the measure t ν e − t dt over R + related to Laguerre polynomials { L ν n } n ≥ , ν > −
1. In fact, letting ω ( t ) = t ν e − t , t > Z ∞ P n ( θ ) P m ( θ ) { ϕ ( t ) } t ν e − t dt = δ n , m . ( . ) The corresponding integral (1.11) is calculated in [1], Vol. II, and we obtain Ω ( x ) = Z ∞ e − x / t − t t ν − dt = x ν / K ν (cid:0) √ x (cid:1) ≡ ρ ν ( x ) , x > , ( . ) where K ν ( z ) is the modified Bessel function or Macdonald function [9]. The function ρ ν has the Mellin-Barnes integral representation in the form (cf. [10]) ρ ν ( x ) = π i Z γ + i ∞γ − i ∞ Γ ( ν + s ) Γ ( s ) x − s ds , x , γ ∈ R + , ν ∈ R , ( . ) where Γ ( z ) is Euler’s gamma-function [1], Vol. I. The asymptotic behavior of the modified Bessel function atinfinity and near the origin [1], Vol. II gives the corresponding values for the function ρ ν , ν ∈ R . Precisely,we have ρ ν ( x ) = O (cid:16) x ( ν −| ν | ) / (cid:17) , x → , ν = , ρ ( x ) = O ( log x ) , x → , ( . ) ρ ν ( x ) = O (cid:16) x ν / − / e − √ x (cid:17) , x → + ∞ . ( . ) Therefore, if the condition (cf. (1.9)) Z ∞ x k ρ ν ( x ) ψ ( x ) dx < ∞ , k ∈ N ( . ) holds valid, we arrive at the following proposition. Proposition 1.
Let ν > − , ϕ , ψ be nonnegative functions defined on R + which are related by the modi-fied Laplace transform ( . ) . Then under condition ( . ) the finiteness of the integral Z ∞ x k e − x / M ψ ( x ) dx < ∞ , k ∈ N ( . ) for some M > the sequence { P n } n ≥ of orthogonal polynomials with respect to the measure ρ ν ( x ) ψ ( x ) dxover R + is compositionally orthogonal in the sense of Laguerre relatively to the function ϕ , i.e. Z ∞ P n ( θ ) P m ( θ ) { ϕ ( t ) } t ν e − t dt = Z ∞ P n ( x ) P m ( x ) ρ ν ( x ) ψ ( x ) dx = δ n , m . ( . ) Proof.
Since via (2.7) the integral 1 t Z ∞ e − x / t x k ψ ( x ) dx , k ∈ N S. Yakubovich converges uniformly with respect to t ∈ [ / M , M ] , M >
0, the consecutive differentiation under the integralsign is allowed, and we derive, recalling (1.6) P n ( θ ) P m ( θ ) { ϕ ( t ) } = P n ( θ ) P m ( θ ) (cid:26) t Z ∞ e − x / t ψ ( x ) dx (cid:27) = t Z ∞ e − x / t P n ( x ) P m ( x ) ψ ( x ) dx . Then Z ∞ P n ( θ ) P m ( θ ) { ϕ ( t ) } t ν e − t dt = lim M → ∞ Z M / M Z ∞ e − x / t P n ( x ) P m ( x ) ψ ( x ) t ν − e − t dxdt = Z ∞ P n ( x ) P m ( x ) ρ ν ( x ) ψ ( x ) dx = δ n , m , where the latter equality is guaranteed by condition (2.6) and Fubini’s theorem. (cid:3) Remark 1.
Letting ν ≥ , ψ ( x ) = x α , α > − , we find the sequence { P ν , α n } n ≥ of Prudnikov’s orthogonalpolynomials studied in [10] Z ∞ P ν , α n ( x ) P ν , α m ( x ) ρ ν ( x ) x α dx = Γ ( α + ) Z ∞ P n ( θ ) P m ( θ ) { t α } t ν e − t dt = δ n , m . ( . ) Hermite polynomials.
Other interesting case is the Hermite orthogonality with respect to the measure e − t dt over R . Our method will work on the following even extension of the modified Laplace transform(1.4) ϕ ( | t | ) = | t | Z ∞ e − x / | t | ψ ( x ) dx , t ∈ R \{ } . ( . ) Hence when the Parseval identity for the Mellin transform [6] suggests the equality Z ∞ e − x / t − t dtt = π i Z γ + i ∞γ − i ∞ Γ (cid:16) s (cid:17) Γ ( s ) x − s ds , x , γ > . ( . ) Invoking the duplication formula for the gamma function [1], we get the right-hand side of (2.11) in the form14 π i Z γ + i ∞γ − i ∞ Γ (cid:16) s (cid:17) Γ ( s ) x − s ds = π / i Z γ / + i ∞γ / − i ∞ Γ ( s ) Γ (cid:18) + s (cid:19) (cid:16) x (cid:17) − s ds = √ π ρ / , (cid:18) x (cid:19) , ( . ) where by ρ ν , k ( x ) we denote the ultra-exponential weight function introduced in [10] ρ ν , k ( x ) = π i Z γ + i ∞γ − i ∞ Γ k ( s ) Γ ( ν + s ) x − s ds , x , ν , γ > , k ∈ N . ( . ) Proposition 2.
Let t > , x ∈ R , ϕ ( t ) , ψ ( x ) be nonnegative functions which are related by ( . ) . If ψ iseven then under conditions ( . ) and Z ∞ − ∞ | x | k ρ / , (cid:18) x (cid:19) ψ ( x ) dx < ∞ , k ∈ N ( . ) omposition orthogonality and new sequences of orthogonal polynomials and functions 5 the sequence { P n } n ≥ of orthogonal polynomials with respect to the measure π − / ρ / , (cid:0) x / (cid:1) ψ ( x ) dx over R is compositionally orthogonal in the sense of Hermite relatively to the function ϕ ( | t | ) , i.e. Z ∞ − ∞ P n ( θ ) P m ( θ ) { ϕ ( | t | ) } e − t dt = Z ∞ − ∞ P n ( x ) P m ( x ) ρ / , (cid:18) x (cid:19) ψ ( x ) dx √ π = δ n , m . ( . ) Proof.
Indeed, since Z ∞ − ∞ P n ( θ ) P m ( θ ) { ϕ ( | t | ) } e − t dt = Z ∞ [ P n ( θ ) P m ( θ ) + P n ( − θ ) P m ( − θ )] { ϕ ( t ) } e − t dt we have due to (1.4), (1.6), (2.7) [ P n ( θ ) P m ( θ ) + P n ( − θ ) P m ( − θ )] { ϕ ( t ) } = t Z ∞ e − x / t [ P n ( x ) P m ( x ) + P n ( − x ) P m ( − x )] ψ ( x ) dx . ( . ) Thus appealing to (2.11), (2.12), (2.14) and Fubini’s theorem, we obtain finally from (2.16) Z ∞ Z ∞ e − x / t − t [ P n ( x ) P m ( x ) + P n ( − x ) P m ( − x )] ψ ( x ) dxdtt = Z ∞ [ P n ( x ) P m ( x ) + P n ( − x ) P m ( − x )] ρ / , (cid:18) x (cid:19) ψ ( x ) dx √ π = Z ∞ − ∞ P n ( x ) P m ( x ) ρ / , (cid:18) x (cid:19) ψ ( x ) dx √ π = δ n , m . (cid:3) Jacobi polynomials.
The modified sequence { P α , β n ( t − ) } n ≥ of these classical polynomials is ortho-gonal with respect to the measure ( − t ) α t β dt , α , β > − [ , ] . Therefore the kernel Ω ( x ) , x > Ω ( x ) = Z ( − t ) α t β − e − x / t dt = e − x Z ∞ t α ( + t ) − α − β − e − xt dt = Γ ( + α ) e − x U ( + α , − β , x ) , ( . ) where U ( a , b , z ) is the Tricomi function [4]. Hence we arrive at Proposition 3.
Let α > − , β > , ϕ , ψ be nonnegative functions defined on R + which are related by themodified Laplace transform ( . ) . Then under the condition Z ∞ x k e − x ψ ( x ) dx < ∞ , k ∈ N ( . ) the sequence { P n } n ≥ of orthogonal polynomials with respect to the measure e − x U ( + α , − β , x ) ψ ( x ) dxover R + is compositionally orthogonal in the sense of Jacobi relatively to the function ϕ , i.e. Z P n ( θ ) P m ( θ ) { ϕ ( t ) } ( − t ) α t β dt = Γ ( + α ) Z ∞ P n ( x ) P m ( x ) e − x U ( + α , − β , x ) ψ ( x ) dx = δ n , m . ( . ) Proof.
Indeed, since P n ( θ ) P m ( θ ) { ϕ ( t ) } = P n ( θ ) P m ( θ ) (cid:26) t Z ∞ e − x / t ψ ( x ) dx (cid:27) = t Z ∞ e − x / t P n ( x ) P m ( x ) ψ ( x ) dx , where the consecutive differentiation under the integral sign is permitted owing to the estimate S. Yakubovich t Z ∞ e − x / t x k ψ ( x ) dx ≤ δ Z ∞ e − x x k ψ ( x ) dx , < δ ≤ t ≤ , k ∈ N , and the latter integral by x is finite via (2.18). Hence, taking into account (2.17), Z P n ( θ ) P m ( θ ) { ϕ ( t ) } ( − t ) α t β dt = lim δ → + Z δ P n ( θ ) P m ( θ ) { ϕ ( t ) } ( − t ) α t β dt = lim δ → + Z δ Z ∞ e − x / t P n ( x ) P m ( x ) ψ ( x )( − t ) α t β − dxdt = Z Z ∞ e − x / t P n ( x ) P m ( x ) ψ ( x )( − t ) α t β − dxdt = Γ ( + α ) Z ∞ P n ( x ) P m ( x ) e − x U ( + α , − β , x ) ψ ( x ) dx = δ n , m , where the interchange of the order of integration is possible by Fubini’s theorem due to (2.18) and an ele-mentary estimate of the Tricomi function Γ ( + α ) U ( + α , − β , x ) = Z ∞ t α ( + t ) − α − β − e − xt dt ≤ Z ∞ t α ( + t ) − α − β − dt = B ( + α , β ) , where B ( a , b ) is the Euler beta function [1], Vol. I. (cid:3)
3. P
ROPERTIES OF P RUDNIKOV ’ S WEIGHT FUNCTIONS AND THEIR PRODUCTS
In this section we will exhibit properties of the weight functions ρ ν + ( x ) ρ ν ( x ) , ρ ν ( x ) , x >
0, where ρ ν ( x ) is defined by (2.2), (2.3), in order to involve them in the sequel to investigate the corresponding orthogonaland multiple orthogonal polynomials. In particular, we will establish their differential properties, integralrepresentations and differential equations. Concerning the function ρ ν , we found in [10] the followingintegral representation in terms of Laguerre polynomials ( − ) n x n n ! ρ ν ( x ) = Z ∞ t ν + n − e − t − x / t L ν n ( t ) dt , n ∈ N . ( . ) It has a relationship with the Riemann-Liouville fractional integral [9] (cid:0) I α − f (cid:1) ( x ) ≡ (cid:0) I α − f ( x ) (cid:1) = Γ ( α ) Z ∞ x ( t − x ) α − f ( t ) dt , Re α > , ( . ) namely, we get the formula ρ ν ( x ) = (cid:0) I ν − ρ (cid:1) ( x ) , ν > . ( . ) Further, the index law for fractional integrals immediately implies ρ ν + µ ( x ) = (cid:0) I ν − ρ µ (cid:1) ( x ) = (cid:0) I µ − ρ ν (cid:1) ( x ) . ( . ) The corresponding definition of the fractional derivative presumes the relation D µ − = − DI − µ − . Hence forthe ordinary n -th derivative of ρ ν we find D n ρ ν ( x ) = ( − ) n ρ ν − n ( x ) , n ∈ N . ( . ) The function ρ ν possesses the following recurrence relation (see [10]) omposition orthogonality and new sequences of orthogonal polynomials and functions 7 ρ ν + ( x ) = νρ ν ( x ) + x ρ ν − ( x ) , ν ∈ R . ( . ) In the operator form it can be written as follows ρ ν + ( x ) = ( ν − xD ) ρ ν ( x ) . ( . ) Now we will derive the Mellin-Barnes representations for the product of functions ρ ν + ρ ν and for thesquare ρ ν . In fact, it can be done, using the related formulas for the product of Macdonald functions. Thus,appealing to Entries 8.4.23.31, 8.4.23.27 in [5], Vol. III, we obtain ρ ν + ( x ) ρ ν ( x ) = − ν i √ π Z γ + i ∞γ − i ∞ Γ ( s + ν + ) Γ ( s + ν ) Γ ( s ) Γ ( s + ν + / ) ( x ) − s ds , x , γ > , ( . ) ρ ν ( x ) = − ν i √ π Z γ + i ∞γ − i ∞ Γ ( s + ν ) Γ ( s + ν ) Γ ( s ) Γ ( s + ν + / ) ( x ) − s ds , x , γ > . ( . ) Immediate consequences of these formulas are relationships of the products of Prudnikov’s weight functions ρ ν + ρ ν , ρ ν with ρ ν + , ρ ν , correspondingly. Indeed, it can be done via Entry 8.4.2.3 in [5], Vol. III andthe Parseval equality for the Mellin transform. Hence we deduce from (3.8), (3.9), respectively, ρ ν + ( x ) ρ ν ( x ) = − ν Z ( − t ) − / t ν − ρ ν + (cid:18) xt (cid:19) dt = x ν √ πΓ ( ν + / ) Z ( − t ) ν − / t − ν − ρ ν + (cid:18) xt (cid:19) dt , ( . ) ρ ν ( x ) = − ν Z ( − t ) − / t ν − ρ ν (cid:18) xt (cid:19) dt = x ν √ πΓ ( ν + / ) Z ( − t ) ν − / t − ν − ρ ν (cid:18) xt (cid:19) dt . ( . ) An ordinary differential equation for the function ρ ν + ρ ν is given by Proposition 4.
The function u ν = ρ ν + ρ ν satisfies the following third order differential equationx d u ν dx + x ( − ν ) d u ν dx + ( ν ( ν − ) − x ) du ν dx + ( ν − ) u ν ( x ) = . ( . ) Proof.
Differentiating u ν with the use of (3.5), we have du ν dx = − ρ ν ( x ) − ρ ν + ( x ) ρ ν − ( x ) . ( . ) Multiplying both sides of (3.13) by x and employing the recurrence relation (3.6), we get x du ν dx = − x ρ ν ( x ) + νρ ν + ( x ) ρ ν ( x ) − ρ ν + ( x ) . Differentiating both sides of the latter equality, using (3.6), (3.13) and the notation u ν = ρ ν + ρ ν , we obtain ddx (cid:18) x du ν dx (cid:19) = − ( + ν ) ρ ν ( x ) + ν du ν dx + u ν ( x ) . ( . ) Differentiating (3.14) and multiplying the result by x , we recall (3.6) to find S. Yakubovich x d dx (cid:18) x du ν dx (cid:19) = − ν ( + ν ) ρ ν ( x ) + ν x d u ν dx + x du ν dx + ( + ν ) u ν ( x ) . ( . ) Hence, expressing ( + ν ) ρ ν ( x ) from (3.14) and fulfilling the differentiation on the left-hand side of (3.15),we get (3.12). (cid:3) Concerning the differential equation for the function ρ ν , we prove Proposition 5.
The function h ν = ρ ν satisfies the following third order differential equationx d h ν dx + x ( − ν ) d h ν dx + ( ν + − ν − x ) dh ν dx + ( ν − ) h ν ( x ) = . ( . ) Proof.
Since x (cid:0) ρ ν ( x ) (cid:1) ′ = − x ρ ν − ( x ) ρ ν ( x ) = νρ ν ( x ) − ρ ν + ( x ) ρ ν ( x ) = ν h ν ( x ) − ρ ν + ( x ) ρ ν ( x ) wederive, owing to (3.5), (3.6) x ddx (cid:18) x dh ν dx − ν h ν ( x ) (cid:19) = − x ddx ( ρ ν + ( x ) ρ ν ( x ))= x h ν ( x ) − νρ ν + ( x ) ρ ν ( x ) + ρ ν + ( x ) = (cid:0) x − ν (cid:1) h ν ( x ) + ν x dh ν dx + ρ ν + ( x ) . Hence one more differentiation yields ddx (cid:18) x ddx (cid:18) x dh ν dx − ν h ν ( x ) (cid:19)(cid:19) = ddx (cid:18) (cid:0) x − ν (cid:1) h ν ( x ) + ν x dh ν dx (cid:19) − ρ ν + ( x ) ρ ν ( x ) . But from the beginning of the proof we find2 ρ ν + ( x ) ρ ν ( x ) = ν h ν ( x ) − x dh ν dx . ( . ) Therefore, substituting this expression into the previous equality, we fulfil the differentiation to arrive at(3.16) and to complete the proof. (cid:3)
Corollary 1 . The following recurrence relations between functions u ν , h ν holdu ν = ν h ν + xu ν − , ( . ) h ν + = ν h ν + x ν u ν − + x h ν − = ν u ν + x h ν − − ν h ν . ( . ) Proof.
Equality (3.18) is a direct consequence of (3.17) and (3.5). Equalities (3.19), in turn, are obtained,taking squares of both sides of (3.6) and employing (3.18). (cid:3) omposition orthogonality and new sequences of orthogonal polynomials and functions 9
4. O
RTHOGONAL POLYNOMIALS WITH ρ ν WEIGHT FUNCTION
The object of this section is to characterize the sequence of orthogonal polynomials { P n } n ≥ , satisfyingthe orthogonality conditions Z ∞ P m ( x ) P n ( x ) ρ ν ( x ) dx = δ n , m , ν > − . ( . ) Clearly, up to a normalization factor conditions (4.1) are equivalent to the equalities Z ∞ P n ( x ) ρ ν ( x ) x m dx = , m = , , . . . , n − , n ∈ N . ( . ) The moments of the weight ρ ν ( x ) can be obtained immediately from the Mellin-Barnes representation (3.9),treating it as the inverse Mellin transform [6]. Hence we get Z ∞ ρ ν ( x ) x µ dx = √ π Γ ( + µ + ν ) Γ ( + µ + ν ) Γ ( + µ ) + ( ν + µ ) Γ ( µ + ν + / ) , µ > max {− , − − ν , − ν } . ( . ) Furthermore, the sequence { P n } n ≥ satisfies the 3-term recurrence relation in the form xP n ( x ) = A n + P n + ( x ) + B n P n ( x ) + A n P n − ( x ) , ( . ) where P ν − ( x ) ≡ , P n ( x ) = ∑ nk = a n , k x k , a n , n = A n + = a n a n + , B n = b n a n − b n + a n + , a n ≡ a n , n , b n ≡ a n , n − . Basing on properties of the functions ρ ν , ρ ν + ρ ν and orthonormality (4.1), we establish Proposition 6 . Let ν > − . The following formulas hold Z ∞ P n ( x ) ρ ν + ( x ) ρ ν ( x ) dx = + ν + n , ( . ) Z ∞ P n ( x ) x ρ ν ( x ) ρ ν − ( x ) dx = + n , ( . ) Z ∞ P n ( x ) x ρ ν ( x ) ρ ν − ( x ) dx = B n − ( ν − ) (cid:18) + n (cid:19) , ( . ) Z ∞ P n ( x ) ρ ν + ( x ) ρ ν ( x ) dx = B n + ( ν + ) (cid:18) + ν + n (cid:19) . ( . ) Proof.
In fact, recurrence relation (3.6) and orthonormality (4.1) implies Z ∞ P n ( x ) ρ ν + ( x ) ρ ν ( x ) dx = ν + Z ∞ P n ( x ) x ρ ν ( x ) ρ ν − ( x ) dx . ( . ) Hence, integrating by parts in the latter integral and eliminating integrated terms by virtue of the asymptoticbehavior (2.4), (2.5) , we find Z ∞ P n ( x ) x ρ ν ( x ) ρ ν − ( x ) dx = Z ∞ (cid:18) P n ( x ) + xP n ( x ) P ′ n ( x ) (cid:19) ρ ν ( x ) dx . ( . ) Therefore, we have from (4.9), (4.10) and orthogonality (4.1) Z ∞ P n ( x ) ρ ν + ( x ) ρ ν ( x ) dx = + ν + n , which proves (4.5). Concerning equality (4.6), it is a direct consequence of (3.6), (4.1). The same idea is toprove (4.7), (4.8), employing (4.3) as well. (cid:3) The composition orthogonality which is associated with the sequence { P n } n ≥ is given by Theorem 1.
Let ν > − / . The sequence { P n } n ≥ is compositionally orthogonal in the sense of Laguerrerelatively to the function t ν e t Γ ( − ν , t ) , where Γ ( µ , z ) is the incomplete gamma function, i.e. Z ∞ t ν e − t P n ( θ ) P m ( θ ) (cid:0) t ν e t Γ ( − ν , t ) (cid:1) dt = δ n , m Γ ( + ν ) . ( . ) Proof.
Writing P n explicitly, we appeal to the integral representation (3.1) for ρ ν ( x ) to write the left-handside of (4.2) in the form Z ∞ P n ( x ) ρ ν ( x ) x m dx = ( − ) m m ! n ∑ k = a n , k ( − ) k k ! Z ∞ Z ∞ t ν + k − e − t − x / t L ν k ( t ) dt × Z ∞ y ν + m − e − y − x / y L ν m ( y ) dydx . ( . ) The Fubini theorem allows to interchange the order of integration in (4.12) due to the convergence of thefollowing integral for all r , r ∈ N Z ∞ Z ∞ t ν + k + r − e − t − x / t dt Z ∞ y ν + m + r − e − y − x / y dy dx = Z ∞ Z ∞ t ν + k + r y ν + m + r e − t − y dtdyt + y ≤ Z ∞ Z ∞ t ν + k + r − / y ν + m + r − / e − t − y dtdy = Γ ( ν + k + r + / ) Γ ( ν + m + r + / ) . Therefore after integration with respect to x we find from (4.12) Z ∞ P n ( x ) ρ ν ( x ) x m dx = ( − ) m m ! n ∑ k = a n , k ( − ) k k ! Z ∞ Z ∞ t ν + k y ν + m e − t − y L ν k ( t ) L ν m ( y ) dtdyt + y . ( . ) However the Rodrigues formula for Laguerre polynomials and Viskov-type identity (1.3) suggest to writethe right-hand side of (4.13) as follows ( − ) m m ! n ∑ k = a n , k ( − ) k k ! Z ∞ Z ∞ t ν + k y ν + m e − t − y L ν k ( t ) L ν m ( y ) dtdyt + y = ( − ) m n ∑ k = a n , k ( − ) k k ! Z ∞ t ν + k e − t L ν k ( t ) Z ∞ θ m (cid:0) y ν e − y (cid:1) dydtt + y . ( . ) The inner integral with respect to y on the right-hand side of the latter equality can be treated via integrationby parts, and we obtain Z ∞ θ m (cid:0) y ν e − y (cid:1) dyt + y = ( − ) m Z ∞ y ν e − y θ m (cid:18) t + y (cid:19) dy . ( . ) omposition orthogonality and new sequences of orthogonal polynomials and functions 11 Working out the differentiation, we find θ m (cid:18) t + y (cid:19) = y m d m dy m m ∑ k = (cid:18) mk (cid:19) ( − ) m − k t m − k ( t + y ) k − ! = ( − ) m t m y m d m dy m (cid:18) t + y (cid:19) = m ! t m y m ( t + y ) m + . Consequently, combining with (4.14), (4.15), equality (4.13) becomes Z ∞ P n ( x ) ρ ν ( x ) x m dx = m ! n ∑ k = a n , k ( − ) k k ! Z ∞ t ν + k + m e − t L ν k ( t ) Z ∞ y ν + m e − y ( t + y ) m + dydt . ( . ) Meanwhile, the integral with respect to y in (4.16) is calculated in (2.17) in terms of the Tricomi function upto a simple change of variables. Thus we get Z ∞ P n ( x ) ρ ν ( x ) x m dx = m ! Γ ( ν + m + ) n ∑ k = a n , k ( − ) k k ! Z ∞ t ν + k + m e − t L ν k ( t ) U ( ν + m + , + ν , t ) dt . ( . ) But appealing to the differential formula 13.3.24 in [4] for the Tricomi function, we have m ! Γ ( ν + m + ) t ν + m U ( ν + m + , + ν , t ) = Γ ( ν + ) θ m ( t ν U ( + ν , + ν , t ))= Γ ( ν + ) θ m (cid:0) t ν e t Γ ( − ν , t ) (cid:1) , ( . ) where Γ ( a , z ) is the incomplete gamma function Γ ( a , z ) = Z ∞ z e − u u a − du . ( . ) Hence, plugging this result in (4.17), recalling the Rodrigues formula for Laguerre polynomials and inte-grating by parts on its right-hand side, we combine with (4.2) to end up with equalities Z ∞ P n ( x ) ρ ν ( x ) x m dx = Γ ( ν + ) × Z ∞ t ν e − t P n ( θ ) θ m (cid:0) t ν e t Γ ( − ν , t ) (cid:1) dt = , m = , , . . . , n − , n ∈ N , ( . ) which yield (4.11) and complete the proof of Theorem 1. (cid:3) Further, the right-hand side of the first equality in (4.20) can be rewritten via integration by parts asfollows Z ∞ t ν + m e − t L ν m ( t ) P n ( θ ) (cid:0) t ν e t Γ ( − ν , t ) (cid:1) dt = , m = , , . . . , n − , n ∈ N . ( . ) Then we expand the function F n ( t ) = θ n ( t ν e t Γ ( − ν , t )) in a series of Laguerre polynomials F n ( t ) = ∞ ∑ r = c n , r L ν r ( t ) , ( . ) where the coefficients c n , r are calculated in terms of the F -hypergeometric functions at unity with the aidof Entry 3.31.12.1 in [2]. Precisely, we get via (4.18) c n , r = r ! Γ ( + ν + r ) Z ∞ t ν e − t L ν r ( t ) θ n (cid:0) t ν e t Γ ( − ν , t ) (cid:1) dt = r ! n ! ( + ν ) n Γ ( + ν + r ) Z ∞ t ν + n e − t L ν r ( t ) U ( ν + n + , + ν , t ) dt = ( + ν ) n Γ ( + ν ) (cid:20) n ! r ! Γ ( ν )( + ν ) r F ( n + , r + , ν + n +
1; 1 − ν ,
1; 1 )+ Γ ( ν + n + ) Γ ( − ν ) F ( ν + r + , ν + n + , ν + n +
1; 1 + ν , + ν ; 1 )] , ( . ) where ( a ) z is the Pochhammer symbol [1], Vol. I and it is valid for nonnegative integers ν by continuity.Hence after substitution the series (4.22) into (4.21) these orthogonality conditions take the form Z ∞ t ν + m e − t L ν m ( t ) n ∑ k = a n , k m ∑ r = c k , r L ν r ( t ) dt = , m = , , . . . , n − , n ∈ N . ( . ) Calculating the integral in (4.24) via relation (2.19.14.8) in [5], Vol. II we obtain d m , r = Z ∞ t ν + m e − t L ν m ( t ) L ν r ( t ) dt = ( − ) r r ! ( + ν ) r Γ ( + ν + m ) × F ( − r , ν + m + , m +
1; 1 + ν ,
1; 1 ) . ( . ) Hence, equalities (4.24) become the linear system of n algebraic equations with n + n ∑ k = a n , k f k , m = , m = , , . . . , n − , n ∈ N , ( . ) where f k , m = m ∑ r = c k , r d m , r . ( . ) Consequently, explicit values of the coefficients a n , k , k = , , . . . , n can be expressed via Cramer’s rule interms of the free coefficient a n , as follows a n , k = − a n , D n , k D n , k = , . . . , n , ( . ) where D n = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f , f , . . . . . . f n , f , . . . . . . . . . f n , . . . . . . . . . . . . . . . ... . . . . . . . . . ... f , n − . . . . . . . . . f n , n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , ( . ) omposition orthogonality and new sequences of orthogonal polynomials and functions 13 D n , k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f , . . . f k − , f , f k + , . . . f n , f , . . . f k − , f , f k + , . . . f n , . . . . . . . . . . . . . . . . . . . . . ... . . . . . . ... . . . . . . ...... . . . . . . ... . . . . . . ...... . . . . . . ... . . . . . . ... f , n − . . . f k − , n − f , n − f k + , . . . f n , n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . ( . ) The free coefficient can be determined, in turn, from the orthogonality conditions (4.1), (4.2), which implythe formula Z ∞ P n ( x ) ρ ν ( x ) x n dx = a n , n . ( . ) Therefore from (4.28) and (4.3) we derive1 a n , n = − a n , √ π D n Γ ( + ν ) Γ ( + ν ) + ν Γ ( ν + / ) n ∑ k = D n , k ( n + k ) ! ( + ν ) n + k ( + ν ) n + k n + k ( ν + / ) n + k , ( . ) where D n , ≡ − D n . Hence a n , = ± D n [ D n , n ] / " √ π Γ ( + ν ) Γ ( + ν ) + ν Γ ( ν + / ) n ∑ k = D n , k ( n + k ) ! ( + ν ) n + k ( + ν ) n + k n + k ( ν + / ) n + k − / , ( . ) where the sign can be chosen accordingly, making positive expressions under the square roots. Assumingalso the positivity of the leading coefficient a n , n we have its value, correspondingly, a n , n = ∓ [ D n , n ] / " √ π Γ ( + ν ) Γ ( + ν ) + ν Γ ( ν + / ) n ∑ k = D n , k ( n + k ) ! ( + ν ) n + k ( + ν ) n + k n + k ( ν + / ) n + k − / . ( . ) Theorem 2 . Let ν > − / . The sequence of orthogonal polynomials { P n } n ≥ can be expressed explicitly,where the coefficients a n , k , k = , , . . . , n are calculated by relations ( . ) and the free term a n , is definedby the equality ( . ) . Besides, it satisfies the 3-term recurrence relation ( . ) , whereA n + = a n , D n + D n , n a n + , D n D n + , n + , B n = D n , n − D n , n − D n + , n D n + , n + . ( . ) An analog of the Rodrigues formula for the sequence { P n } n ∈ N can be established, appealing to the integralrepresentation of an arbitrary polynomial in terms of the associated polynomial of degree 2 n (see [10]).Hence, employing (2.2), we deduce P n ( x ) = ρ ν ( x ) Z ∞ t ν − e − t − x / t q n ( t ) dt = ρ ν ( x ) Z ∞ u ν − e − u − x / u du Z ∞ t ν − e − t − x / t q n ( t ) dt = ( − ) n ρ ν ( x ) d n dx n Z ∞ Z ∞ ( ut ) ν + n − e − u − t − x ( / u + / t ) q n ( t ) dtdu ( u + t ) n , ( . ) where q n ( x ) = n ∑ k = a n , k ( − ) k k ! x k L ν k ( x ) ( . ) and the k -th differentiation under the integral sign is permitted due to the estimate Z ∞ Z ∞ ( ut ) ν + k − e − u − t − x ( / u + / t ) | q n ( t ) | dtdu ( u + t ) k ≤ − k Z ∞ u ν + k / − e − u du Z ∞ t ν + k / − e − t | q n ( t ) | dt = − k Γ (cid:18) ν + k (cid:19) Z ∞ t ν + k / − e − t | q n ( t ) | dt < ∞ , k = , . . . , n . Then writing ( ut ) n ( u + t ) n = ( n − ) ! Z ∞ e − ( / u + / t ) y y n − dy , we get from (4.36) P n ( x ) = ( − ) n ( n − ) ! ρ ν ( x ) d n dx n Z ∞ Z ∞ Z ∞ y n − ( ut ) ν − e − u − t − ( x + y ) / u − ( x + y ) / t q n ( t ) dtdudy = ( − ) n ( n − ) ! ρ ν ( x ) d n dx n Z ∞ Z ∞ y n − t ν − e − t − ( x + y ) / t q n ( t ) ρ ν ( x + y ) dtdy . ( . ) Then, expressing q n in terms of the Laguerre polynomials q n ( x ) = n ∑ k = h n , k L ν k ( x ) , ( . ) where the coefficients h n , k are calculated by virtue of (4.28), (4.37) and relation (2.19.14.15) in [5], Vol. II,namely, h n , k = k ! Γ ( + ν + k ) Z ∞ t ν e − t L ν k ( t ) q n ( t ) dt = − k ! a n , D n Γ ( + ν + k ) n ∑ r = D n , r ( − ) r r ! Z ∞ t ν + r e − t L ν k ( t ) L ν r ( t ) dt = − a n , D n n ∑ r = D n , r r ! ( + ν ) r F ( − k , + ν + r , + r ; 1 + ν ,
1; 1 ) . So, h n , k = − a n , D n n ∑ r = D n , r r ! ( + ν ) r F ( − k , + ν + r , + r ; 1 + ν ,
1; 1 ) , ( . ) and the values of the generalized hypergeometric function can be simplified via relations (7.4.4; 90,91,92,93)in [5], Vol. III. Precisely, we get for k = , . . . , n F ( − k , + ν + r , + r ; 1 + ν ,
1; 1 ) = , k > r , ( . ) omposition orthogonality and new sequences of orthogonal polynomials and functions 15 ( + ν ) r F ( − k , + ν + r , + r ; 1 + ν ,
1; 1 ) = ( r ) ! r ! , k = r , ( + ν ) r F ( − k , + ν + r , + r ; 1 + ν ,
1; 1 ) = − ( r − ) ! ( ν + r )( r − ) ! , k = r − , ( + ν ) r F ( − k , + ν + r , + r ; 1 + ν ,
1; 1 ) = ( ( r − )) !2 r ! (cid:0) r ( r + ν − )( r − )+ r ( r − )( r + ν − )( r + ν )) , k = ( r − ) . Therefore, returning to (4.38) and minding (4.39), (4.40), (4.41), (3.1), (3.2) we find P n ( x ) = ( − ) n + a n , D n ( n − ) ! ρ ν ( x ) d n dx n n ∑ k = n ∑ r = D n , r r ! ( + ν ) r F ( − k , + ν + r , + r ; 1 + ν ,
1; 1 ) × Z ∞ Z ∞ y n − t ν − e − t − ( x + y ) / t L ν k ( t ) ρ ν ( x + y ) dtdy = ( − ) n + a n , D n ( n − ) ! ρ ν ( x ) d n dx n n ∑ k = k ! n ∑ r = D n , r r ! ( + ν ) r F ( − k , + ν + r , + r ; 1 + ν ,
1; 1 ) × Z ∞ y n − d k dx k (cid:16) ( x + y ) k ρ ν ( x + y ) (cid:17) ρ ν ( x + y ) dy = ( − ) n + a n , D n ( n − ) ! ρ ν ( x ) d n dx n n ∑ k = k ! n ∑ r = D n , r r ! ( + ν ) r F ( − k , + ν + r , + r ; 1 + ν ,
1; 1 ) × Z ∞ x ( y − x ) n − d k dy k (cid:16) y k ρ ν ( y ) (cid:17) ρ ν ( y ) dy = − a n , D n ρ ν ( x ) n ∑ k = d k dx k (cid:16) x k ρ ν ( x ) (cid:17) n ∑ r = D n , r r ! × ( + ν ) r k ! F ( − k , + ν + r , + r ; 1 + ν ,
1; 1 )= − a n , D n ρ ν ( x ) n ∑ r = D n , r r ! ( + ν ) r r ∑ k = d k dx k (cid:16) x k ρ ν ( x ) (cid:17) k ! F ( − k , + ν + r , + r ; 1 + ν ,
1; 1 ) . Thus it proves
Theorem 3 . Let ν > − / , n ∈ N . Orthogonal polynomials P n satisfy the Rodrigues-type formulaP n ( x ) = − a n , D n ρ ν ( x ) n ∑ r = D n , r r ! ( + ν ) r r ∑ k = k ! d k dx k (cid:16) x k ρ ν ( x ) (cid:17) F ( − k , + ν + r , + r ; 1 + ν ,
1; 1 ) , ( . ) where a n , is defined by ( . ) and D n , D n , r by ( . ) , ( . ) , respectively. Corollary 2 . Orthogonal polynomials P n have the following representationP n ( x ) = − a n , D n n ∑ r = D n , r r ! ( + ν ) r " r ∑ k = A k , k − ( x )( k ) ! F ( − k , + ν + r , + r ; 1 + ν ,
1; 1 ) + r − ∑ k = A k , k ( x )( k + ) ! F ( − k − , + ν + r , + r ; 1 + ν ,
1; 1 ) , ( . ) where A k , k − , A k , k are the type multiple orthogonal polynomials of degree k, associated with the vector ofweight functions ( ρ ν , ρ ν + ) . Proof.
In fact, we write (4.42) in the form P n ( x ) = − a n , D n ρ ν ( x ) n ∑ r = D n , r r ! ( + ν ) r " r ∑ k = ( k ) ! d k dx k (cid:16) x k ρ ν ( x ) (cid:17) F ( − k , + ν + r , + r ; 1 + ν ,
1; 1 )+ r − ∑ k = ( k + ) ! d k + dx k + (cid:16) x k + ρ ν ( x ) (cid:17) F ( − k − , + ν + r , + r ; 1 + ν ,
1; 1 ) . ( . ) Meanwhile, appealing to the Rodrigues formulas for the type 1 multiple orthogonal polynomials associatedwith the vector of weight functions ( ρ ν , ρ ν + ) (see in [7]), it gives d k dx k (cid:16) x k ρ ν ( x ) (cid:17) = A k , k − ( x ) ρ ν ( x ) + B k , k − ( x ) ρ ν + ( x ) , d k + dx k + (cid:16) x k + ρ ν ( x ) (cid:17) = A k , k ( x ) ρ ν ( x ) + B k , k ( x ) ρ ν + ( x ) , where A k , k − , B k , k − are polynomials of degree k , k −
1, respectively, and A k , k , B k , k are polynomials of degree k . Therefore, substituting these expressions into (4.44), we obtain P n ( x ) = − a n , D n n ∑ r = D n , r r ! ( + ν ) r " r ∑ k = A k , k − ( x )( k ) ! F ( − k , + ν + r , + r ; 1 + ν ,
1; 1 )+ r − ∑ k = A k , k ( x )( k + ) ! F ( − k − , + ν + r , + r ; 1 + ν ,
1; 1 ) − a n , ρ ν + ( x ) D n ρ ν ( x ) n ∑ r = D n , r r ! ( + ν ) r " r ∑ k = B k , k − ( x )( k ) ! F ( − k , + ν + r , + r ; 1 + ν ,
1; 1 )+ r − ∑ k = B k , k ( x )( k + ) ! F ( − k − , + ν + r , + r ; 1 + ν ,
1; 1 ) . But the existence of a multiple orthogonal polynomial sequence with respect to the vector of weight functions ( ρ ν , ρ ν + ) implies the identity r ∑ k = B k , k − ( x )( k ) ! F ( − k , + ν + r , + r ; 1 + ν ,
1; 1 )+ r − ∑ k = B k , k ( x )( k + ) ! F ( − k − , + ν + r , + r ; 1 + ν ,
1; 1 ) ≡ , which drives to (4.43) and completes the proof. (cid:3) omposition orthogonality and new sequences of orthogonal polynomials and functions 17 Remark 2 . In a similar manner orthogonal polynomials with the weight function ρ ν + ( x ) ρ ν ( x ) can beinvestigated. We leave this topic to the interested reader.Finally, in this section we establish the generating function for polynomials P n , which is defined as usuallyby the equality G ( x , z ) = ∞ ∑ n = P n ( x ) z n n ! , x > , z ∈ C , ( . ) where | z | < h x and h x > G ( x , z ) = ρ ν ( x ) ∞ ∑ n = z n n ! n ∑ k = h n , k k ! d k dx k h x k ρ ν ( x ) i = ρ ν ( x ) ∞ ∑ n = z n n ! n ∑ k = h n , k k ∑ j = (cid:18) kj (cid:19) ( − ) j ( k − j ) ! x j ρ ν − j ( x ) . Meanwhile, the product x j ρ ν − j ( x ) is expressed in [3] as follows x j ρ ν − j ( x ) = x j / r j ( √ x ; ν ) ρ ν ( x ) + x ( j − ) / r j − ( √ x ; ν − ) ρ ν + ( x ) , j ∈ N , where r − ( z ; ν ) = x j / r j ( √ x ; ν ) = ( − ) j [ j / ] ∑ i = ( ν + i − j + ) j − i ( j − i + ) i x i i ! . ( . ) Therefore this leads to the final expression of the generating function for the sequence ( P n ) n ∈ N , namely, G ( x , z ) = ∞ ∑ n = n ∑ k = k ∑ j = (cid:18) kj (cid:19) ( − ) j h n , k n ! ( k − j ) ! x j / r j ( √ x ; ν ) z n + ρ ν + ( x ) ρ ν ( x ) ∞ ∑ n = n ∑ k = k ∑ j = (cid:18) kj (cid:19) ( − ) j h n , k n ! ( k − j ) ! x ( j − ) / r j − ( √ x ; ν − ) z n , ( . ) where coefficients h n , k are defined by (4.40).5. N OTE ON THE MULTIPLE ORTHOGONAL POLYNOMIALS
In this section we will exhibit two types of multiple orthogonal polynomials for the vector of weightfunctions ( ρ ν , ρ ν + , ρ ν ρ ν + ) , ν > − / R + with an additional factor x α , α > −
1. Precisely, weconsider the type 1 polynomials ( A α n , B α n − , C α n − ) , n ∈ N of degree n , n −
1, respectively, satisfying theorthogonality conditions Z ∞ (cid:2) A α n ( x ) ρ ν ( x ) + B α n − ( x ) ρ ν + ( x ) + C α n − ( x ) ρ ν ( x ) ρ ν + ( x ) (cid:3) x α + m dx = , m = , , . . . , n − . ( . ) So, we have 3 n linear homogeneous equations with 3 n + A α n , B α n − , C α n − .Therefore we can find type 1 polynomials up to a multiplicative factor. Let us denote the function q α n , n − , n − for the convenience q α n , n − , n − ( x ) = A α n ( x ) ρ ν ( x ) + B α n − ( x ) ρ ν + ( x ) + C α n − ( x ) ρ ν ( x ) ρ ν + ( x ) . ( . ) Type 2 polynomials p α n + , n , n are monic polynomials of degree 3 n + Z ∞ p α n + , n , n ( x ) ρ ν ( x ) x α + m dx = , m = , , . . . , n , ( . ) Z ∞ p α n + , n , n ( x ) ρ ν + ( x ) x α + m dx = , m = , , . . . , n − , ( . ) Z ∞ p α n + , n , n ( x ) ρ ν ( x ) ρ ν + ( x ) x α + m dx = , m = , , . . . , n − . ( . ) This gives 3 n + n + | ν | < / Theorem 4 . Let n , m , l ∈ N , | ν | < / , f n , g m , h l be polynomials of degree at most n , m , l, respectively.Let f n ( x ) ρ ν ( x ) + g m ( x ) ρ ν + ( x ) + h l ( x ) ρ ν ( x ) ρ ν + ( x ) = ( . ) for all x > . Then f n ≡ , g m ≡ , h l ≡ . Proof.
As is known (cf. [10]) the quotient ρ ν / ρ ν + is represented by the Ismail integral ρ ν ( x ) ρ ν + ( x ) = π Z ∞ s − ds ( x + s )( J ν + ( √ s ) + Y ν + ( √ s )) , ( . ) where J ν , Y ν are Bessel functions of the first and second kind, respectively [5], Vol. II. In fact, let r ≥ max { n , m + , l + } . Hence, dividing (5.6) by ρ ν ( x ) ρ ν + ( x ) and using (3.6), we find f n ( x ) ρ ν ( x ) ρ ν + ( x ) + xg m ( x ) ρ ν − ( x ) ρ ν ( x ) + ν g m ( x ) + h l ( x ) = . ( . ) Then, differentiating r times, it gives d r dx r (cid:20) f n ( x ) ρ ν ( x ) ρ ν + ( x ) (cid:21) + d r dx r (cid:20) xg m ( x ) ρ ν − ( x ) ρ ν ( x ) (cid:21) = . ( . ) Assuming f n ( x ) = ∑ nk = f n , k x k and employing (5.7), the first term on the left-hand side of (5.9) can be treatedas follows d r dx r (cid:20) f n ( x ) ρ ν ( x ) ρ ν + ( x ) (cid:21) = π d r dx r n ∑ k = f n , k x k Z ∞ e − xy dy Z ∞ e − sy s − dsJ ν + ( √ s ) + Y ν + ( √ s )= π n ∑ k = f n , k ( − ) k d r dx r Z ∞ d k dy k (cid:2) e − xy (cid:3) dy Z ∞ e − sy s − dsJ ν + ( √ s ) + Y ν + ( √ s )= π n ∑ k = f n , k ( − ) k Z ∞ ∂ k + r ∂ y k ∂ x r (cid:2) e − xy (cid:3) dy Z ∞ e − sy s − dsJ ν + ( √ s ) + Y ν + ( √ s )= π n ∑ k = f n , k ( − ) k + r Z ∞ d k dy k (cid:2) y r e − xy (cid:3) dy Z ∞ e − sy s − dsJ ν + ( √ s ) + Y ν + ( √ s ) , where the differentiation under the integral sign is possible via the absolute and uniform convergence. Now,we integrate k times by parts in the outer integral with respect to y on the right-hand side of the latter equality,eliminating the integrated terms due to the choice of r , and then differentiate under the integral sign in theinner integral with respect to s owing to the same arguments, to obtain omposition orthogonality and new sequences of orthogonal polynomials and functions 19 d r dx r (cid:20) f n ( x ) ρ ν ( x ) ρ ν + ( x ) (cid:21) = π Z ∞ y r e − xy Z ∞ e − sy s − J ν + ( √ s ) + Y ν + ( √ s ) n ∑ k = f n , k ( − ) k + r s k ! ds . ( . ) In the same fashion the second term in (5.9) is worked out to find ( g m ( x ) = ∑ mk = g m , k x k ) d r dx r (cid:20) xg m ( x ) ρ ν − ( x ) ρ ν ( x ) (cid:21) = π Z ∞ y r e − xy Z ∞ e − sy J ν ( √ s ) + Y ν ( √ s ) m ∑ k = g m , k ( − ) k + r + s k ! ds . ( . ) Substituting (5.10), (5.11) into (5.9) and cancelling twice the Laplace transform via its injectivity for inte-grable and continuous functions [6], we arrive at the equality xg m ( − x ) (cid:2) J ν + ( √ x ) + Y ν + ( √ x ) (cid:3) + f n ( − x ) (cid:2) J ν ( √ x ) + Y ν ( √ x ) (cid:3) = , x > . ( . ) The sum of squares of Bessel functions in brackets is called the Nicholson kernel, which has the Mellin-Barnes representation by virtue of Entry 8.4.20.35 in [5], Vol. III x k (cid:2) J ν ( √ x ) + Y ν ( √ x ) (cid:3) = − k cos ( πν ) π / i Z γ + i ∞γ − i ∞ Γ ( s + k ) Γ ( s + k + ν ) Γ ( s + k − ν ) × Γ (cid:18) − s − k (cid:19) ( x ) − s ds , | ν | − k < γ < − k . ( . ) Then, using on the right-hand side of (5.13) the reflection formula for gamma function [1], Vol. I, it can bewritten as follows2 − k cos ( πν ) π / i Z γ + i ∞γ − i ∞ Γ ( s + k ) Γ ( s + k + ν ) Γ ( s + k − ν ) Γ (cid:18) − s − k (cid:19) ( x ) − s ds = − k ( − ) k cos ( πν ) π / i Z γ + i ∞γ − i ∞ Γ ( s + k ) Γ ( s + k + ν ) Γ ( s + k − ν ) Γ ( / − s )( / + s ) k ( x ) − s ds . ( . ) Our goal now is to shift the contour to the right to make integration along the straight line with | ν | < Re s < /
2. To do this we should take into account the residues at k simple poles s m = − / − m , m = , , . . . , k − s = s m (cid:18) Γ ( s + k ) Γ ( s + k + ν ) Γ ( s + k − ν ) Γ ( / − s )( / + s )( / + s ) . . . ( s + k − / ) ( x ) − s (cid:19) = Γ ( s m + k ) Γ ( s m + k + ν ) Γ ( s m + k − ν ) Γ ( / − s m )( / + s m )( / + s m + ) . . . ( / + s m + m − )( / + s m + m + ) . . . ( / + s m + k − )( x ) − s m = ( − ) m ( x ) / + m ( k − m − ) ! Γ ( k − m − / ) Γ ( k − m − / + ν ) Γ ( k − m − / − ν ) . Therefore we get from (5.14)2 − k cos ( πν ) π / i Z γ + i ∞γ − i ∞ Γ ( s + k ) Γ ( s + k + ν ) Γ ( s + k − ν ) Γ (cid:18) − s − k (cid:19) ( x ) − s ds = − k ( − ) k cos ( πν ) π / i Z µ + i ∞µ − i ∞ Γ ( s + k ) Γ ( s + k + ν ) Γ ( s + k − ν ) Γ ( / − s )( / + s ) k ( x ) − s ds − − k √ x cos ( πν ) π / k − ∑ m = ( − ) k + m ( x ) m ( k − m − ) ! Γ ( k − m − / ) Γ ( k − m − / + ν ) Γ ( k − m − / − ν ) , ( . ) where | ν | < µ < /
2. Now, recalling Parseval’s equality for the Mellin transform and Entries 8.4.2.5,8.4.23.27 in [5], Vol. III, we derive from (5.13), (5.15) x k (cid:2) J ν ( √ x ) + Y ν ( √ x ) (cid:3) = − k ( − ) k √ x π cos ( πν ) Z ∞ K ν ( √ t ) t k − / x + t dt − − k √ x cos ( πν ) π / k − ∑ m = ( − ) k + m ( x ) m ( k − m − ) ! Γ ( k − m − / ) Γ ( k − m − / + ν ) Γ ( k − m − / − ν ) . ( . ) Analogously, we find x k + (cid:2) J ν + ( √ x ) + Y ν + ( √ x ) (cid:3) = − k ( − ) k √ x π cos ( πν ) Z ∞ K ν + ( √ t ) t k + / x + t dt − − k √ x cos ( πν ) π / k ∑ m = ( − ) k + m + ( x ) m ( k − m ) ! Γ ( k − m + / ) Γ ( k − m + / + ν ) Γ ( k − m + / − ν ) . ( . ) Substituting expressions (5.16), (5.17) in (5.12), we derive after straightforward simplifications Z ∞ K ν + ( √ t ) g m ( t ) √ tx + t dt + Z ∞ K ν ( √ t ) f n ( t ) dt √ t ( x + t )+ √ π m ∑ k = k ∑ j = ( − x ) j k − j ( k − j ) ! Γ ( k − j + / ) Γ ( k − j + / + ν ) Γ ( k − j + / − ν ) −√ π n − ∑ k = k ∑ j = ( − x ) j k − j ( k − j ) ! Γ ( k − j + / ) Γ ( k − j + / + ν ) Γ ( k − j + / − ν ) = , x > . ( . ) Last two terms in (5.18) are polynomials of degree m , n −
1, respectively. Hence, differentiating through r ≥ max { n , m + } times by x , we obtain d r dx r Z ∞ K ν + ( √ t ) g m ( t ) √ tx + t dt + d r dx r Z ∞ K ν ( √ t ) f n ( t ) dt √ t ( x + t ) = . ( . ) The left-hand side of (5.19) is the r -th derivative of the sum of two Stieltjes transforms which are, in turn,two fold Laplace transforms. Consequently, fulfilling the differentiation under the integral sign in (5.19)as above owing to the absolute and uniform convergence by x ≥ x >
0, we cancel Laplace transforms ofintegrable functions via the injectivity. Then with (2.2) it yields the equality f n ( x ) ρ ν ( x ) + g m ( x ) ρ ν + ( x ) = , x > . ( . ) Comparing with (5.6), we see that h l ≡ . Further, identity (5.20) implies that f n , g m have the same positiveroots, if any. Let x > g m . Then dividing (5.20) by g m and making a differentiation, we get − ρ ν ( x ) ρ ν − ( x ) f n ( x ) g m ( x ) + ρ ν ( x ) f ′ n ( x ) g m ( x ) − f n ( x ) g ′ m ( x ) g m ( x ) − ρ ν + ( x ) ρ ν ( x ) = . omposition orthogonality and new sequences of orthogonal polynomials and functions 21 Since ρ ν ( x ) >
0, we divide the previous equation by ρ ν , multiply by x , g m and employ (3.6) to find − g m ( x ) ρ ν + ( x ) ( f n ( x ) + xg m ( x )) + ρ ν ( x ) (cid:0) ν f n ( x ) g m ( x ) + x (cid:0) f ′ n ( x ) g m ( x ) − f n ( x ) g ′ m ( x ) (cid:1)(cid:1) = . ( . ) However, the existence of the type 1 multiple orthogonal polynomials with respect to the vector of weightfunctions ( ρ ν , ρ ν + ) suggests the equalities g m ( x ) ( f n ( x ) + xg m ( x )) ≡ , ν f n ( x ) g m ( x ) + x (cid:0) f ′ n ( x ) g m ( x ) − f n ( x ) g ′ m ( x ) (cid:1) ≡ . ( . ) So, if g m ≡
0, it proves the theorem. Otherwise f n ( x ) + xg m ( x ) ≡
0, and with the second equation in (5.22)we have x ( ν + ) g m ( x ) ≡ . Thus g m ≡ f n ≡ (cid:3) Remark 3 . The choice of ν is important. For instance, for ν = − / f n ( x ) ≡ − x , g m ( x ) ≡ , h l ( x ) ≡ Theorem 5 . Let ν ∈ [ , / ) . For every α > ddx (cid:2) x α q α n , n − , n − ( x ) (cid:3) = x α − q α − n , n − , n ( x ) ( . ) and the following differential recurrence relations holdA α − n ( x ) = ( α + ν ) A α n ( x ) + x [ A α n ( x )] ′ − xC α n − ( x ) , ( . ) B α − n − ( x ) = α B α n − ( x ) + x [ B α n − ( x )] ′ − C α n − ( x ) , ( . ) C α − n ( x ) = ( α + ν ) C α n − ( x ) + x [ C α n − ( x )] ′ − A α n ( x ) − xB α n − ( x ) . ( . ) Proof.
From (5.1), (5.2) and integration by parts we get Z ∞ q α n , n − , n − ( x ) x α + m dx = − m + Z ∞ ddx (cid:2) x α q α n , n − , n − ( x ) (cid:3) x m + dx = , where the integrated terms vanish for every α > − Z ∞ ddx (cid:2) x α q α n , n − , n − ( x ) (cid:3) x m dx = , m = , . . . , n . But, evidently, Z ∞ ddx (cid:2) x α q α n , n − , n − ( x ) (cid:3) dx = , α > . Therefore Z ∞ ddx (cid:2) x α q α n , n − , n − ( x ) (cid:3) x m dx = , m = , . . . , n . ( . ) Now, working out the differentiation in (5.27), involving (3.5), (3.6), we find ddx (cid:2) x α q α n , n − , n − ( x ) (cid:3) = α x α − q α n , n − , n − ( x )+ x α − (cid:0) x [ A α n ( x )] ′ ρ ν ( x ) + x [ B α n − ( x )] ′ ρ ν + ( x ) + x [ C α n − ( x )] ′ ρ ν ( x ) ρ ν + ( x ) + A α n ( x ) ρ ν ( x )( νρ ν ( x ) − ρ ν + ( x )) − xB α n − ( x ) ρ ν + ( x ) ρ ν ( x ) − xC α n − ( x ) ρ ν ( x ) + C α n − ( x ) ρ ν + ( x )( νρ ν ( x ) − ρ ν + ( x )) (cid:1) . Thus ddx (cid:2) x α q α n , n − , n − ( x ) (cid:3) = x α − (cid:2) A α − n ( x ) ρ ν ( x ) + B α − n − ( x ) ρ ν + ( x ) + C α − n ( x ) ρ ν ( x ) ρ ν + ( x ) (cid:3) , ( . ) where A α − n ( x ) , B α − n − ( x ) , C α − n ( x ) are polynomials of degree at most n , n − , n , respectively, being definedby formulas (5.24), (5.25), (5.26). The linear homogeneous system (5.27) of 3 n + n + x α − q α − n , n − , n ( x ) , and the representation (5.28) is unique by virtue of Theorem 4. This proves (5.23) andcompletes the proof of Theorem 5. (cid:3) Remark 4 . The same analysis for the function q α n , n − , n ( x ) does not work. In fact, in this case we deriveanalogously Z ∞ ddx (cid:2) x α q α n , n − , n ( x ) (cid:3) x m dx = , m = , . . . , n + . However, working out the differentiation, we will get polynomials of degree at most n + , n , n , respectively,which implies 3 n + n + p α n + , n , n (or 3-orthogonal polynomials). Theorem 6.
For every ν ≥ , α > − ddx p α n + , n , n ( x ) = ( n + ) p α + n + , n − , n ( x ) . ( . ) Proof.
Recalling (3.5), (3.6) and asymptotic behavior of the weight functions (2.4), (2.5), we integrate byparts in (5.3), eliminating the integrated terms, to deduce Z ∞ ddx (cid:2) p α n + , n , n ( x ) (cid:3) ρ ν ( x ) x α + + m dx = , m = , , . . . , n − . ( . ) Concerning equalities (5.4), (5.5), it corresponds the following ones Z ∞ ddx (cid:2) p α n + , n , n ( x ) (cid:3) ρ ν + ( x ) x α + + m dx = , m = , , . . . , n − , ( . ) Z ∞ ddx (cid:2) p α n + , n , n ( x ) (cid:3) ρ ν ( x ) ρ ν + ( x ) x α + + m dx = , m = , , . . . , n − . ( . ) Now h p α n + , n , n ( x ) i ′ is a polynomial of degree 3 n with leading coefficient 3 n + p α + n + , n − , n ( x ) . Hence we get (5.29) by unicity. Theorem 6 is proved. (cid:3) omposition orthogonality and new sequences of orthogonal polynomials and functions 23 R EFERENCES1. A. Erd´elyi,W. Magnus, F. Oberhettinger, and F.G. Tricomi, Higher Transcendental Functions,Vols. I and II, McGraw-Hill,NewYork, London, Toronto, 1953.2. Yu.A. Brychkov, O.I. Marichev, N.V. Savischenko,