M a t h e m a t i c a B a l k a n i c a New Series Vol. 6,, Fasc. - Integral Representations of Functional Series with Members Containing Jacobi Polynomials Dragana Jankov, Tibor K. Pogány Presented at 6 th International Conference TMSF In this article we establish a double definite integral representation, and two other indefinite integral expressions for a functional series and its derivative with members containing Jacobi polynomials. MSC : Primary 33C45, 4A3; Secondary 6D7, 4C Key Words: Jacobi polynomials, functional series integral representation. Introduction and motivation A family yx = p n x p n x n + p n x n + + p x + p, p n, of polynomials of degree exactly n N := {,,,... } is a family of classical continuous orthogonal polynomials if it is the solution of a differential equation of the type p xy x + p xy x + π n yx =, where p x = ax + bx + c is a polynomial of at most second order and p x = dx+e is a linear polynomial [8, 3]. Since the polynomial p n x has exact degree n, by equating the highest coefficients of x n in one gets π n = ann + dn. The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials, together with their special cases which are ultraspherical polynomials, the Čebyšev polynomials and the Legendre polynomials.
4 D. Jankov, T.K. Pogány In this article, our main aim is to derive integral representations for the functional series with members containing Jacobi polynomials. This will be realized in a similar manner as the authors have done it in the articles on Neumann series [3, 4] and on the Kapteyn series, []. The Jacobi polynomials, which are also called hypergeometric polynomials, can be represented with the following formula [8] P n α,β z = + α [ n n, + α + β + n F z ]. n! + α When α = β =, the polynomial becomes the Legendre polynomial. The Gegenbauer polynomials, and also the Čebyšev polynomials, are special cases of the Jacobi polynomials. From it follows that P n α,β z is a polynomial of degree precisely n and that P n α,β = + α n. n! The Jacobi polynomials are orthogonal with respect to the weight function wx = x α + x β on the interval [, ]. Assurance of the integrability of wx is achieved by requiring α > and β >, see []. The orthogonal polynomials with the weight function b x α x a β, on the finite interval [a, b] can be expressed in the form [] constant P n α,β x a b a. It is worth mentioning that Luke and Wimp [5] proved that if we have continuous function fx, which has a piecewise continuous derivative for x λ, then fx may be expanded into a uniformly convergent series of shifted Jacobi polynomials in the form fx = n= a n λp α,β n x/λ, where ɛ x/λ ɛ, ɛ >, α >, β >. Various techniques are available for the determination of the coefficients a n λ. Let us define a functional series in the following form P α,β z := a n P n α,β z, z C, 3 where a n are constants and P n α,β stands for the Jacobi polynomial. We point out that the Bulgarian mathematician P. Rusev studied in [] the convergence
Integral Representations of Functional Series... 5 of the series P α,β z precisely, he considered a + P α,β z. For that purpose, he used the asymptotic formula by Darboux see [6], [, Eq. 8..9]: P n α,β z = P α,β z n / ω n z + p α,β n z, where ωz is the inverse of Žukovsky transformation z = ω + ω for which ω =, P α,β z and p n α,β z are analytic functions holomorphic n N in the region C \ [, ] and such that lim z = uniformly on every n pα,β n compact subset of this region. Further, for < r < +, he denoted by Er := Intγr, where γr := {z C: ωz = r}; thus ad definitionem, E = C. He obtained the following result written in our present notation. Theorem [, Proposition..] Let η = lim sup n n a n. Then: i if η, the series P α,β z is divergent in the whole region C \ [, ]; ii if η <, the P α,β z is absolutely uniformly convergent on every compact subset of the region Eη and diverges at every point of the region C \ Eη.. Integral representation In this section we will derive the double integral representation for the Rusev series 3. For that purpose, we will replace z C with x R and assume that the behavior of a n n N ensures the convergence of our main series. We would also need some symbols and formulae which we present as follows. By convention, [a] and {a} = a [a] denote the integer and fractional part of some real number a, respectively. The Laplace integral representation for the Dirichlet series, which is given below, following mainly [], [, C. V]: D λ x = a n e λnx = x [λ t] e xt a n dt, where the convention is followed that the real sequence λ n n N monotonically increases and tends to infinity; equivalently < λ < λ < < λ n. Also taking a function x a x = ax, where a C [k, m], k, m Z, k < m, then by using the operator d x := + {x} d dx,
6 D. Jankov, T.K. Pogány we get the following condensed form of the Euler Maclaurin summation formula [7, p. 365]: m m a j = ax + {x}a x m dx = d x axdx. j=k+ k Now, we are ready to formulate the following theorem. Theorem. Let a C R + and a N = a n n N. Then for all α > /, α + β > and for all x of the domain I a := max{, η }, ] 4 we have the integral representation [s] Γs + P s α,β x P α,β x = s Γα + s + Γβ + s + aw Γα + w + d Γβ + w + w ds dw. Γw + k P r o o f. First, we begin by establishing the convergence conditions for the series P α,β x. For that purpose, let us consider the integral representation given by Feldheim [9]: P α,β n x = Γα + β + n + t α+β+n e t L α n xt dt, 5 valid for all n N, α + β >, where L α n is the Laguerre polynomial. We estimate 5 via the bounding inequality for Laguerre functions L µ ν x, given by Love [4, p. 396, Theorem ]: L µ ν x ΓRν + µ + Γν + ΓRµ + ΓRµ + Γµ + e x, 6 where ν C, x >, Rµ > and Rµ + ν >, which has been generalized by Pogány and Srivastava [6]. Specifying µ = α R, ν = n N the bound 6 reduces to L α Γn + α + n x n! Γα + ex, x >. 7 Now, applying bound 7 to the integrand of 5, we have that P α,β x α+β+ a n Γα + n + n. Γα + + x n! + x The resulting power series converges uniformly for all x satisfying constraint 4.
Integral Representations of Functional Series... 7 A more convenient integral representation for the Jacobi polynomials has been given by Braaksma and Meulenbeld [5], [7, p. 9] P n α,β z = n 4 n α + nβ + n πn! zu ± i n z v u α v β dudv, z, where minα, β >. This expression in an obvious way one reduces to P n α,β x = n α + nβ + n πn! Thus, combining 3 and 8 we get P α,β x = π i x u + x v n u α v β dudv, x. 8 where D a u, v is the Dirichlet series D a u, v = u α v β D a u, v dudv, 9 a n α + nβ + n n! e n ln i x u +x v. The Dirichlet series possesses Laplace integral representation when its parameter has positive real part, therefore we are looking for the two-dimensional region S uv x in the uv plane where { R ln i x u + x v } = ln + xv + xu <. So, we get the ellipse S uv x = { u, v R : + xv + xu < / }, such that is nonempty for all x I a, so D a u, v converges in I a. Now, the related Laplace integral and the Euler Maclaurin summation formula see for instance [3], [] give us: D a u, v = ln i x u + x v Γα + Γβ + [s] i x u + x v s aw Γα + w + d Γβ + w + w ds dw. Γw +
8 D. Jankov, T.K. Pogány Substituting into 9 we get P α,β x = πγα + Γβ + Denoting I x s := [s] u α v β ln i x u + x v s i x u + x v aw Γα + w + d Γβ + w + w du dv ds dw. Γw + we get I x sds = ln i x u + x v i x u + x v s u α v β du dv, i x u + x v s u α v β dudv = π Γα + Γβ + α,β Γs + P s x Γα + s + Γβ + s +. Therefore, we can easily conclude that I x s = πγα + Γβ + Γs + P s α,β x s Γα + s + Γβ + s +. Finally, by using and, we immediately get the proof of the theorem, with the assertion that the integration domain R + becomes [, because [s] is equal to zero for all s [,. Remark. In the previous theorem, we used Love s bound [4] for the Laguerre function L µ ν x. Similar results one can get using some other bounds for Laguerre polynomials, i.e. for the Laguerre functions. Let us mention some of them. Pogány and Srivastava [6, p. 354, Theorem ] derived an extension of Love s bounding inequality. The magnitude of their bounds is Ox µ/ c e x, c see [6] which results in a convergence region similar to I a. There are two, well known see, e.g. [], classical global uniform estimates for the Jacobi polynomials, given by Szegő [], subsequently improved by Rooney [9]. However both these bounds, having magnitudes Oe x/, are inferior to Love s and to the one by Pogány and Srivastava in [6, p. 354].
Integral Representations of Functional Series... 9 Other inequalities for the Laguerre functions, that is for the Jacobi function and the Jacobi polynomials can also be found in [4, 6, ]. Asymptotic estimates for the Jacobi functions can be found e.g. in [, ]. 3. Indefinite integral representations for P α,β x In this section, we will deduce another, indefinite type integral representations for the functional series 3, by using the fact that the Jacobi polynomials P α,β n x satisfy the linear homogeneous ODE of the second order [8, ]: x y + β α + α + βx y + n + α + β + ny =. 3 Now, multiplying 3 with a n and then summing up that expression in n N we immediately get the following equality x P α,β x + β α + α + βx P α,β x = a n n + α + β + np n α,β x =: R α,β x, where the right hand side expression R α,β x is the functional series associated with the series P α,β x. In the following theorem, the first main result of this section is given. Theorem. For all α >, α + β > the particular solution of the linear ODE: x y + β α + α + βx y = R α,β x, 4 represents the first derivative x P α,βx of the functional series 3. Here for a C R +, a N = a n n N and letting n a n absolutely converges, for all x I a we have the integral representation R α,β x = [s] s Γs + P s α,β x Γα + s + Γβ + s + aw w + α + β + w Γα + w + d Γβ + w + w ds dw. Γw + P r o o f. Equation 4 was established in the beginning of this section. Further, the uniform convergence of the series R α,β x can be easily recognized, using the convergence conditions of the series P α,β x, to be such that n a n <. Then, using an integral representation derived in Theorem, with a n n + α + β + n a n, we readily get the statement.
D. Jankov, T.K. Pogány Below, we shall introduce another indefinite integral representation for the series P α,β x. have Theorem 3. Let the situation be the same as in Theorem. Then we P α,β x = x α+ + x β+ R α,βx x α + x dx β dx, where R α,β x is the series associated with the series P α,β x. P r o o f. It is easy to see that the Jacobi polynomial P α,β x = is a solution of the homogeneous differential equation x y + β α + α + βx y =. 5 So, a guess of the particular solution is P α,β x = P α,β xwx = wx. Substituting this form into nonhomogeneous differential equation 4, we get x w x + β α + α + βx w x = R α,β x. It is easy to check that the previous equation can be rewritten into [ x α+ + x β+ w x] = Rα,β x x α + x β ; so we have that w x = x α+ + x β+ R α,βx x α + x β dx + C. Finally, the desired particular solution is Rα,β x x α + x β dx wx = x α+ + x β+ dx + x + C α β B ; β, α + C, where Bt; p, q = t p t q dt denotes the so called Čebyšev integral incomplete Beta function. As P α,β is a solution of homogeneous differential equation 5, it does not contribute to the particular solution, so the constants C, C can be taken to be zero and we immediately get the assertion of the theorem. Acknowledgements. The first authors research is partially covered by Grant No 35-3588-39 of Ministry of Science, Education and Sports of the Republic of Croatia.
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D. Jankov, T.K. Pogány [6] T.K. Pogány, H.M. Srivastava, Some improvements over Love s inequality for the Laguerre function, Integral transforms Spec. Funct. 8, No 5 7, 35 358. [7] T.K. Pogány, E. Süli, Integral representation for Neumann series of Bessel functions, Proc. Amer. Math. Soc. 37, No 7 9, 363 368. [8] E.D. Rainville, Special Functions, The Macmillan Company, New York, 96. [9] P.G. Rooney, Further inequalities for generalized Laguerre polynomials, C. R. Math. Rep. Acad. Sci. Canada 7 985, 73 75. [] P. Rusev, Expansion of analytic functions in series of classical orthogonal polynomials, Banach Center Publ. 983, 87 98. [] H.M. Srivastava, Some bounding inequalities for the Jacobi and related functions, Banach J. Math. Anal., No 7, 3 38. [] G. Szegő, Orthogonal Polynomials, American Mathematical Society Colloquium Publ., Vol. 3, Revised ed. American Mathematical Society, Providence, R.I. 959. Department of Mathematics, University of Osijek Trg Ljudevita Gaja 6, Osijek 3, CROATIA e-mail: djankov@mathos.hr Faculty od Maritime Studies, University of Rijeka Studentska, Rijeka 5, CROATIA e-mail: poganj@pfri.hr Received: October 7,