MSC 2010: Primary 33C45, 40A30; Secondary 26D07, 40C10In 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

Jankov, Dragana

Pogany, Tibor K.

Bulgarian Digital Mathematics Library at IMI-BAS

M a t h e m a t i c aB a l k a n i c a—————————NewSeries Vol. 26, 2012, Fasc. 1-2Integral Representations of Functional Serieswith Members Containing Jacobi PolynomialsDragana Jankov 1, Tibor K. Poga´ny 2Presented at 6th International Conference “TMSF’ 2011”In this article we establish a double definite integral representation, and two otherindefinite integral expressions for a functional series and its derivative with members containingJacobi polynomials.MSC 2010: Primary 33C45, 40A30; Secondary 26D07, 40C10Key Words: Jacobi polynomials, functional series’ integral representation1. Introduction and motivationA familyy(x) = pn(x) ≡ pnxn + pn−1xn−1 + · · ·+ p1x+ p0, pn 6= 0,of polynomials of degree exactly n ∈ N0 := {0, 1, 2, . . . } is a family of classicalcontinuous orthogonal polynomials if it is the solution of a differential equationof the typep2(x)y′′(x) + p1(x)y′(x) + piny(x) = 0 , (1)where p2(x) = ax2+ bx+ c is a polynomial of at most second order and p1(x) =dx+e is a linear polynomial [8, 13]. Since the polynomial pn(x) has exact degreen, by equating the highest coefficients of xn in (1) one getspin = − (an(n− 1) + dn) .The most widely used orthogonal polynomials are the classical orthogonal poly-nomials, consisting of the Hermite polynomials, the Laguerre polynomials andthe Jacobi polynomials, together with their special cases which are ultrasphericalpolynomials, the Cˇebysˇev polynomials and the Legendre polynomials.104 D. Jankov, T.K. Poga´nyIn this article, our main aim is to derive integral representations forthe functional series with members containing Jacobi polynomials. This willbe realized in a similar manner as the authors have done it in the articles onNeumann series [3, 4] and on the Kapteyn series, [2].The Jacobi polynomials, which are also called hypergeometric polynomi-als, can be represented with the following formula [18]P (α,β)n (z) =(1 + α)nn! 2F1[ −n, 1 + α+ β + n1 + α∣∣∣1− z2]. (2)When α = β = 0, the polynomial (2) becomes the Legendre polynomial. TheGegenbauer polynomials, and also the Cˇebysˇev polynomials, are special casesof the Jacobi polynomials. From (2) it follows that P (α,β)n (z) is a polynomial ofdegree precisely n and thatP (α,β)n (1) =(1 + α)nn!.The Jacobi polynomials are orthogonal with respect to the weight functionw(x) = (1 − x)α(1 + x)β on the interval [−1, 1]. Assurance of the integrabilityof w(x) is achieved by requiring α > −1 and β > −1, see [22].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 [22]constant · P (α,β)n(2x− ab− a − 1).It is worth mentioning that Luke and Wimp [15] proved that if we have contin-uous function f(x), which has a piecewise continuous derivative for 0 ≤ x ≤ λ,then f(x) may be expanded into a uniformly convergent series of shifted Jacobipolynomials in the formf(x) =∞∑n=0an(λ)P (α,β)n (2x/λ− 1) ,where ² ≤ x/λ ≤ 1− ², ² > 0, α > −1, β > −1. Various techniques are availablefor the determination of the coefficients an(λ).Let us define a functional series in the following formPα,β(z) :=∞∑n=1anP(α,β)n (z), z ∈ C, (3)where an are constants and P(α,β)n stands for the Jacobi polynomial. We pointout that the Bulgarian mathematician P. Rusev studied in [20] the convergenceIntegral Representations of Functional Series . . . 105of the series Pα,β(z) (precisely, he considered a0 +Pα,β(z)). For that purpose,he used the asymptotic formula by Darboux (see [6], [22, Eq. (8.21.9)]):P (α,β)n (z) = P(α,β)(z)n−1/2 ωn(z) (1 + p(α,β)n (z)) ,where ω(z) is the inverse of Zˇukovsky transformation z = 12(ω+ω−1) for whichω(∞) =∞, P (α,β)(z) 6= 0 and (p(α,β)n (z))n∈N are analytic functions holomorphicin the region C \ [−1, 1] and such that limn→∞ p(α,β)n (z) = 0 uniformly on everycompact subset of this region. Further, for 1 < r < +∞, he denoted by E(r) :=Intγ(r), where γ(r) := {z ∈ C : |ω(z)| = r}; thus ad definitionem, E(∞) = C.He obtained the following result (written in our present notation).Theorem [20, Proposition 1.1.] Let η = lim supn→∞ n√|an|. Then:(i) if η ≥ 1, the series Pα,β(z) is divergent in the whole region C \ [−1, 1];(ii) if 0 ≤ η < 1, the Pα,β(z) is absolutely uniformly convergent on everycompact subset of the region E(η−1) and diverges at every point of theregion C \E(η−1).2. Integral representationIn this section we will derive the double integral representation for theRusev series (3). For that purpose, we will replace z ∈ C with x ∈ R and assumethat the behavior of (an)n∈N ensures the convergence of our main series.We would also need some symbols and formulae which we present asfollows. By convention, [a] and {a} = a − [a] denote the integer and fractionalpart of some real number a, respectively.The Laplace integral representation for the Dirichlet series, which is givenbelow, following mainly [10], [12, C. §V]:Dλ(x) =∞∑n=1ane−λnx = x∫ ∞0e−xt([λ−1(t)]∑n=1an)dt ,where the convention is followed that the real sequence(λn)n∈N monotonicallyincreases and tends to infinity; equivalently0 < λ1 < λ2 < · · · < λn ↑ ∞ .Also taking a function x 7→ ax = a(x), where a ∈ C1[k,m], k,m ∈ Z, k < m,then by using the operatordx := 1 + {x} ddx ,106 D. Jankov, T.K. Poga´nywe get the following condensed form of the Euler–Maclaurin summation formula[17, p. 2365]:m∑j=k+1aj =∫ mk(a(x) + {x}a′(x))dx = ∫ mkdxa(x)dx .Now, we are ready to formulate the following theorem.Theorem 1. Let a ∈ C1(R+) and a|N = (an)n∈N. Then for all α >−1/2, α+ β > −1 and for all x of the domainIa :=(max{0, 2η − 1}, 1] (4)we have the integral representationPα,β(x) = −∫ ∞1∫ [s]0∂∂s(Γ(2s+ 1)P (α,β)s (x)Γ(α+ s+ 12)Γ(β + s+12))× dw(a(w) Γ(α+ w + 12)Γ(β + w +12)Γ(2w + 1))dsdw .P r o o f. First, we begin by establishing the convergence conditions forthe series Pα,β(x). For that purpose, let us consider the integral representationgiven by Feldheim [9]:P (α,β)n (x) =1Γ(α+ β + n+ 1)∫ ∞0tα+β+ne−t L(α)n(12(1− x)t)dt , (5)valid for all n ∈ N0, α + β > −1, where L(α)n is the Laguerre polynomial. Weestimate (5) via the bounding inequality for Laguerre functions L(µ)ν (x), givenby Love [14, p. 396, Theorem 2]:|L(µ)ν (x)| ≤Γ(<(ν + µ+ 1))|Γ(ν + 1)|Γ(<(µ) + 1)Γ(<(µ) + 12)|Γ(µ+ 12)|ex , (6)where ν ∈ C, x > 0, <(µ) > −12 and <(µ+ν) > −1, which has been generalizedby Poga´ny and Srivastava [16]. Specifying µ = α ∈ R, ν = n ∈ N0 the bound(6) reduces to|L(α)n (x)| ≤Γ(n+ α+ 1)n! Γ(α+ 1)ex, x > 0 . (7)Now, applying bound (7) to the integrand of (5), we have that|Pα,β(x)| ≤ 1Γ(α+ 1)(21 + x)α+β+1 ∞∑n=1|an|Γ(α+ n+ 1)n!( 21 + x)n.The resulting power series converges uniformly for all x satisfying constraint (4).Integral Representations of Functional Series . . . 107A more convenient integral representation for the Jacobi polynomials hasbeen given by Braaksma and Meulenbeld [5], [7, p. 191]P (α,β)n (1− 2z2) =(−1)n4n(α+ 12)n(β + 12)npi(2n)!∫ 1−1∫ 1−1(zu± i√1− z2 v)2n× (1− u2)α−12 (1− v2)β−12 dudv, 0 ≤ z ≤ 1,where 2min(α, β) > −1. This expression in an obvious way one reduces toP (α,β)n (x) =2n(α+ 12)n(β +12)npi(2n)!∫ 1−1∫ 1−1(i√1− xu−√1 + x v)2n× (1− u2)α−12 (1− v2)β−12 dudv, |x| ≤ 1. (8)Thus, combining (3) and (8) we getPα,β(x) =1pi∫ 1−1∫ 1−1(1− u2)α−12 (1− v2)β−12 Da(u, v) dudv, (9)where Da(u, v) is the Dirichlet seriesDa(u, v) =∞∑n=1an (α+ 12)n(β +12)n(2n)!e−n ln(√2(i√1−xu−√1+x v))−2.The Dirichlet series possesses Laplace integral representation when its parameterhas positive real part, therefore we are looking for the two-dimensional regionSuv(x) in the uv–plane where<{ln 2(i√1− xu−√1 + x v)2} = ln 2((1 + x)v2 + (1− x)u2) < 0 .So, we get the ellipseSuv(x) ={(u, v) ∈ R2 : (1 + x)v2 + (1− x)u2 < 1/2} ,such that is nonempty for all x ∈ Ia, so Da(u, v) converges in Ia.Now, the related Laplace–integral and the Euler–Maclaurin summationformula (see for instance [3], [2]) give us:Da(u, v) = −ln(√2(i√1− xu−√1 + x v))2Γ(α+ 12)Γ(β +12)×∫ ∞0∫ [s]0(√2(i√1− xu−√1 + x v))2s× dw(a(w) Γ(α+ w + 12)Γ(β + w +12)Γ(2w + 1))dsdw. (10)108 D. Jankov, T.K. Poga´nySubstituting (10) into (9) we getPα,β(x) = − 1piΓ(α+ 12)Γ(β +12)∫ 1−1∫ 1−1∫ ∞0∫ [s]0(1− u2)α−12 (1− v2)β−12× ln (√2(i√1− xu−√1 + x v))2 · (√2(i√1− xu−√1 + x v))2s× dw(a(w) Γ(α+ w + 12)Γ(β + w +12)Γ(2w + 1))dudv dsdw . (11)DenotingIx(s) :=∫ 1−1∫ 1−1ln(√2(i√1− xu−√1 + x v))2× (√2(i√1− xu−√1 + x v))2s (1− u2)α−12 (1− v2)β−12 dudv,we get∫Ix(s)ds =∫ 1−1∫ 1−1(√2(i√1− xu−√1 + x v))2s(1− u2)α−12 (1− v2)β−12dudv= piΓ(α+ 12)Γ(β +12)Γ(2s+ 1)P(α,β)s (x)Γ(α+ s+ 12)Γ(β + s+12).Therefore, we can easily conclude thatIx(s) = piΓ(α+ 12)Γ(β + 12)∂∂s(Γ(2s+ 1)P (α,β)s (x)Γ(α+ s+ 12)Γ(β + s+12)). (12)Finally, by using (11) and (12), we immediately get the proof of the theorem,with the assertion that the integration domain R+ becomes [1,∞) because [s]is equal to zero for all s ∈ [0, 1).Remark 2. In the previous theorem, we used Love’s bound [14] for theLaguerre function L(µ)ν (x). Similar results one can get using some other boundsfor Laguerre polynomials, i.e. for the Laguerre functions. Let us mention someof them.Poga´ny and Srivastava [16, p. 354, Theorem 2] derived an extension ofLove’s bounding inequality. The magnitude of their bounds is O(x−µ/2−cex)),c ≥ 0 (see [16]) which results in a convergence region similar to Ia.There are two, well–known (see, e.g. [1]), classical global uniform esti-mates for the Jacobi polynomials, given by Szego˝ [22], subsequently improvedby Rooney [19]. However both these bounds, having magnitudes O(ex/2), areinferior to Love’s and to the one by Poga´ny and Srivastava in [16, p. 354].Integral Representations of Functional Series . . . 109Other inequalities for the Laguerre functions, that is for the Jacobi func-tion and the Jacobi polynomials can also be found in [14, 16, 21]. Asymptoticestimates for the Jacobi functions can be found e.g. in [1, 11].3. Indefinite integral representations for Pα,β(x)In this section, we will deduce another, indefinite type integral represen-tations for the functional series (3), by using the fact that the Jacobi polynomialsP(α,β)n (x) satisfy the linear homogeneous ODE of the second order [18, 22]:(1− x2)y′′ + (β − α− (2 + α+ β)x) y′ + n(1 + α+ β + n)y = 0 . (13)Now, multiplying (13) with an and then summing up that expression in n ∈ Nwe immediately get the following equality(1− x2)P′′α,β(x) + (β − α− (2 + α+ β)x)P′α,β(x)= −∞∑n=1an n(1 + α+ β + n)P (α,β)n (x) =: Rα,β(x) ,where the right–hand side expression Rα,β(x) is the functional series associatedwith the series Pα,β(x). In the following theorem, the first main result of thissection is given.Theorem 2. For all α > −12 , α+ β > −1 the particular solution of thelinear ODE:(1− x2)y′ + (β − α− (2 + α+ β)x) y = Rα,β(x) , (14)represents the first derivative ∂∂xPα,β(x) of the functional series (3). Here fora ∈ C1(R+), a|N = (an)n∈N and letting∑∞n=1 n2an absolutely converges, for allx ∈ Ia we have the integral representationRα,β(x) =∫ ∞1∫ [s]0∂∂s(Γ(2s+ 1)P (α,β)s (x)Γ(α+ s+ 12)Γ(β + s+12))× dw(a(w)w (1 + α+ β + w) Γ(α+ w + 12)Γ(β + w +12)Γ(2w + 1))dsdw .P r o o f. Equation (14) was established in the beginning of this section.Further, the uniform convergence of the series Rα,β(x) can be easily recog-nized, using the convergence conditions of the series Pα,β(x), to be such that∑∞n=1 n2|an| < ∞. Then, using an integral representation derived in Theorem1, with an 7→ −n (1 + α+ β + n) an, we readily get the statement.110 D. Jankov, T.K. Poga´nyBelow, we shall introduce another indefinite integral representation forthe series Pα,β(x).Theorem 3. Let the situation be the same as in Theorem 2. Then wehavePα,β(x) =∫1(1− x)α+1(1 + x)β+1(∫Rα,β(x)(1− x)α(1 + x)β dx)dx ,where Rα,β(x) is the series associated with the series Pα,β(x).P r o o f. It is easy to see that the Jacobi polynomial P (α,β)0 (x) = 1 is asolution of the homogeneous differential equation(1− x2)y′′ + (β − α− (2 + α+ β)x) y′ = 0 . (15)So, a guess of the particular solution is Pα,β(x) = P(α,β)0 (x)w(x) = w(x). Sub-stituting this form into nonhomogeneous differential equation (14), we get(1− x2)w′′(x) + (β − α− (2 + α+ β)x)w′(x) = Rα,β(x) .It is easy to check that the previous equation can be rewritten into[(1− x)α+1(1 + x)β+1w′(x)]′= Rα,β(x)(1− x)α(1 + x)β ;so we have thatw′(x) =1(1− x)α+1(1 + x)β+1(∫Rα,β(x)(1− x)α(1 + x)β dx+ C1).Finally, the desired particular solution isw(x) =∫ ∫Rα,β(x)(1− x)α(1 + x)β dx(1− x)α+1(1 + x)β+1 dx+ C1 2−α−β−1B(1 + x2;−β,−α)+ C2 ,where B(t; p, q) =∫tp−1(1 − t)q−1dt denotes the so–called Cˇebysˇev integral(incomplete Beta function).As P (α,β)0 is a solution of homogeneous differential equation (15), it doesnot contribute to the particular solution, so the constants C1, C2 can be takento be zero and we immediately get the assertion of the theorem.Acknowledgements. The first authors research is partially covered byGrant No 235-2352818-1039 of Ministry of Science, Education and Sports of theRepublic of Croatia.Integral Representations of Functional Series . . . 111References[1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover,New York (1964).[2] A´. Baricz, D. Jankov, T.K. Poga´ny, Integral representation of first kindKapteyn series, J. Math. Phys. 52, No 4 (2011), 043518, 7pp.[3] A´. Baricz, D. Jankov, T.K. Poga´ny, Integral representations for Neumann-type series of Bessel functions Iν , Yν and Kν , Proc. Amer. Math. Soc. 140,No 3 (2012), 951–960.[4] A´. Baricz, D. Jankov, T.K. Poga´ny, Neumann series of Bessel functions,Integral Transforms Spec. Funct. (2012), To appear.[5] B.L.J. Braaksma, B. Meulenbeld, Jacobi polynomials as spherical harmo-nics, Nederl. Akad. Wetensch. Proc. Ser. A 71 = Indag. Math. 30 (1968),384–389.[6] G. Darboux, Me´moire sur l’approximation des fonctions de tre`s grandnombres et sur une classe e´tendue de de´veloppments en se´rie, Journal deMathe´matiques, 3e serie IV (1878), 5–56, 377–416.[7] A. Dijksma, T.H. Koornwinder, Spherical harmonics and the product oftwo Jacobi polynomials, Nederl. Akad. Wetensch. Proc. Ser. A 74 = Indag.Math. 33 (1971), 191–196.[8] W.N. Everitt, K.H. Kwon, L.L. Littlejohn, R. Wellman, Orthogonal poly-nomial solutions of linear ordinary differential equations, J. Comput. Appl.Math. 133 (2001), 85–109.[9] E. Feldheim, Relations entre les polynoˆmes de Jacobi, Laguerre et Hermite,Acta. Math. 74 (1941), 117–138.[10] G.H. Hardy, M. Riesz, The General Theory of Dirichlet’s Series, UniversityPress, Cambridge (1915).[11] E. Hille, On Laguerre’s series, Firs note. Proc. Nat. Acad. Sci. 11 (1926),261–269.[12] J. 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Rooney, Further inequalities for generalized Laguerre polynomials, C.R. Math. Rep. Acad. Sci. Canada 7 (1985), 273–275.[20] P. Rusev, Expansion of analytic functions in series of classical orthogonalpolynomials, Banach Center Publ. 11 (1983), 287–298.[21] H.M. Srivastava, Some bounding inequalities for the Jacobi and relatedfunctions, Banach J. Math. Anal. 1, No 1 (2007), 131–138.[22] G. Szego˝, Orthogonal Polynomials, American Mathematical Society Collo-quium Publ., Vol. 23, Revised ed. American Mathematical Society, Provi-dence, R.I. (1959).1 Department of Mathematics, University of OsijekTrg Ljudevita Gaja 6, Osijek – 31000, CROATIAe-mail: djankov@mathos.hr2 Faculty od Maritime Studies, University of RijekaStudentska 2, Rijeka – 51000, CROATIAe-mail: poganj@pfri.hr Received: October 17, 2011

Integral Representations of Functional Series with Members Containing Jacobi Polynomials

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Integral Representations of Functional Series with Members Containing Jacobi Polynomials

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