2010 Mathematics Subject Classification: 33C45, 40G05.In this paper we give some results concerning the equiconvergence and equisummability of series in Jacobi polynomials

Boychev, Georgi

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Serdica Math. J. 37 (2011), 323-330EQUICONVERGENCE AND EQUISUMMABILITYOF JACOBI SERIESGeorgi BoychevCommunicated by O. MushkarovAbstract. In this paper we give some results concerning the equiconver-gence and equisummability of series in Jacobi polynomials.1. Jacobi polynomials and series in the complex plane. Letα > −1 and β > −1. The polynomials{P(α,β)n (z)}+∞n=0defined by the equalitiesP (α,β)n (z) =(n+ αn)F(−n, n+ α+ β + 1, α+ 1; 1− z2),z ∈ C, n = 0, 1, 2, . . . ,where F (a, b, c; ζ) is the Gauss hypergeometric function, are called Jacobi poly-nomials with parameters α and β.2010 Mathematics Subject Classification: 33C45, 40G05.Key words: Jacobi polynomials, Jacobi series, asymptotic formula, convergence, summabiliy,equisummabiliy, equiconvergence.324 Georgi BoychevThere exists a unique complex function h, holomorhpic in the region G =C\[−1,+1], such that h2(z) = z2− 1 when z ∈ G and h(x) > 0 when x > 1. Thevalue of this function at any point z ∈ G is denoted by √z2 − 1. The function wdefined in G asw(z) = z +√z2 − 1is holomorphic in G. Moreover, w(z) 6= 0 and [w(z) + (w(z))−1]/2 = z when z ∈G. Therefore, w(z) may be considered as an inverse of the Zhukovsky function.As it is well known, the latter one is univalent in the region D = {w : |w(z)| > 1}and maps it onto G. Hence, the function w maps G onto D. The function w ismeromorphic in the region G = C\[−1,+1] and limz→∞w(z) =∞.G. Szego¨ gives an asymptotic expansion for Jacobi polynomials in theregion G [4, Theorem 8.21.1]. From this expansion it follows the asymptoticformula (n ≥ 1, p = 0, 1, 2, . . . )(1.1) P (α,β)n (z) = ψ(z)n−1/2(w(z))n{p∑k=0ϕk(z)n−k + ϕn,p(z)},where ψ(z) 6= 0, {ϕk(z)}+∞k=0, {ϕn,p(z)}+∞n=1 (p = 0, 1, 2, . . . ) are a complex func-tions holomorphic in the region G and(1.2) ϕn,p(z) = O(n−p−1) (n→∞)uniformly on every compact subset of G. In particular, ϕ0(z) ≡ 1.If p = 0, then it follows from (1.1) and (1.2) that [3, (II.3.1)](1.3) P (α,β)n (z) = ψ(z)n−1/2(w(z))n{1 + ϕn,0(z)},where ϕn,0(z) = O(n−1) (n→∞) uniformly on every compact subset of G.Suppose that r > 1. Denote by γ(r) the image of the circle C(0, r) ={z ∈ C : z = r exp iθ, 0 ≤ θ ≤ 2pi} by the Zhukovsky transformation. As it iswell known, γ(r) is the ellipse with focuses at the points −1 and +1 and semiaxes(r + r−1)/2 and (r − r−1)/2.Let E(r) be the interior of γ(r). Further, we denote E(∞) = C, E(1) = ∅,E∗(r) = C\E(r) (1 < r <∞), E∗(∞) = ∅ and E∗(1) = G.The series of the kind(1.4)+∞∑n=0anP(α,β)n (z)Equiconvergence and equisummability of Jacobi series 325has the name Jacobi series.Theorem 1.1 [3, (IV.1.1), (a)]. If the series (1.4) converges at a pointz0 ∈ G, then it is absolutely uniformly convergent on every compact subset of theregion E(r) with r = |w(z0)|.We remark that in the proof of this theorem it is used the asymptoticformula (1.3).Theorem 1.2 [3, (IV.1.1), (b)]. If(1.5) r = max{1,(limn→+∞sup |an|1/n)−1},then the series (1.4) is absolutely uniformly convergent on every compact subsetof the domain E(r) and diverges in E∗(r).The equality (1.5) can be regarded as a formula of Cauchy-Hadamardtype for the Jacobi series. If r > 1, then the sum of the series (1.4) is a complexfunction, holomorphic in the domain E(r).2. (C, δ)-summability of a series. Let us remind that a series(2.1)+∞∑n=0unis said to be (C, δ)-summable for δ > −1, if there existsσ = limn→+∞(Aδn)−1n∑k=0Aδ−1n−ksk,where Aδk = Γ(k + δ + 1)/[Γ(k + 1)Γ(δ + 1)] (δ > −1; k = 0, 1, . . . ) and {sk}+∞k=0are the partial sums of the series (2.1).Usually the complex number σ is called (C, δ)-sum of the series (2.1).Every convergent series is (C, δ)-summable for every δ > 0 and its (C, δ)-sum isequal to its sum in the usual sense.326 Georgi Boychev3. Equisummability of Jacobi series. Let us to consider a seriesof kind(3.1)+∞∑n=1ann−1/2(w(z))n.In the following proposition we give a relation between the series (1.4) and (2.1).Theorem 3.1. Let δ > −1 and z0 ∈ G. Then the series (1.4) is (C, δ)-summable at z0 if and only if the series (2.1) is (C, δ)-summable at z0.P r o o f. Let p be an integer such that p > δ+1. Further, we assume thata0 = 0.Suppose that the series (1.4) is (C, δ)-summable for z = z0. Applying [2,Theorem 46], we obtain that(3.2) anP(α,β)n (z0) = o(nδ) (n→∞).We define the numbers µ1, µ2, . . . , µp by the following equalitiesµ1 = ϕ1(z0)µ2 = ϕ2(z0)− µ1ϕ1(z0). . . . . . . . . . . . . . . . . . . . . . . . . . . . .µp = ϕp(z0)−p−1∑j=1ϕp−j(z0)µj.Ifbn = anP(α,β)n (z0){1− µ1n−1 − · · · − µpn−p},then from [2, Theorem 74] it follows that the series+∞∑k=1bn is (C, δ)-summable.Using the asymptotic formulas (1.1) and (1.2), we conclude thatbn = ψ(z0)ann−1/2(w(z0))n{1 + dn,p(z0)},where dn,p(z0) = O(n−p−1) (n → ∞). Obviously, ψ(z0) 6= 0 and P (α,β)n (z0) 6= 0(n = 1, 2, . . . ) Then, from (3.2), (1.1) and (1.2) it follows thatanψ(z0)n−1/2(w(z0))n = o(nδ) (n→∞).Equiconvergence and equisummability of Jacobi series 327Further, we haveqn = ψ(z0)ann−1/2(w(z0))ndn,p(z0) = o(nδ)O(n−p−1) (n→∞).Since p > δ + 1 thenqn = O(n−2) (n→∞).Hence, the series+∞∑k=1qn is convergent. Applying [3, Theorem 45] we de-duce that this series is (C, δ)-summable. Then, from the equalitiesann−1/2(w(z0))n = bn/ψ(z0)− qn (n = 1, 2, . . . )it follows that the series (3.1) is (C, δ)-summable.Now, let us assume that the series (3.1) is (C, δ)-summable for z = z0.Thenann−1/2(w(z0))n = o(nδ) (n→∞).This asymptotic formula and the asymptotic formula (1.2) yieldcn = ψ(z0)ann−1/2(w(z0))ndn,p(z0) = o(nδ)O(n−p−1).Since p > δ + 1, then cn = O(n−2). Hence the series+∞∑k=1cn converges.According to [3, Theorem 45] it is (C, δ)-summable.The series of the kind+∞∑n=1an(w(z0))nn−s−1/2,where s = 1, 2, . . . , p, is (C, δ)-summable by Theorem 45 from [3]. Then, fromthe equalities (n = 1, 2, . . . )anP(α,β)n (z0) = ψ(z0)ann−1/2(w(z0))n + ψ(z0)p∑s=1an(w(z0))nϕs(z0)n−s−1/2 + cnit follows that the series (3.1) is (C, δ)-summable for z = z0. Thus Theorem 3.1is proved. Having in mind the proof of Theorem 3.1 it is easy to establish the fol-lowing result:328 Georgi BoychevTheorem 3.2. If δ > −1 and K is a compact subset of the region G,then the series (1.4) is uniformly (C, δ)-summable on K if and only if the series(3.1) is uniformly (C, δ)-summable on K.The following statement is a corollary of Theorem 3.1.Theorem 3.3. If z0 ∈ G, then the series (1.4) converges for z = z0 ifand only if the series (3.1) converges for z = z0.Theorems 3.1 and 3.3 can be called Theorems for equisummability andequiconvergence of Jacobi series, respectively.4. Fatou-Riesz theorem for Jacobi series. Let(4.1)+∞∑n=0antnbe a power series with finite and different from zero radius r of convergenceand g(t) be a sum of (4.1) in the region int{C(0, r)}. It is well known that ifthe coefficients {an}+∞n=0 satisfy certain additional conditions, there is a relationbetween the regular points of the function g(t) on the circumference C(0, r) andthe behaviour of (4.1) on the same circumference. The following result holds [6,(9.21)].Theorem 4.1 (Fatou-Riesz). If the power series (4.1) has a finite radiusr of convergence and anrn = o(nδ) (n → +∞) for some δ > −1, then the series(4.1) is uniformly (C, δ)-summable on every close arc of C(0, r), all points ofwhich are regular for the function f(t).V. Yatsun in [5] described the relations between the singular points ofJacobi series (1.5) and those of the corresponding power series (4.1). In particularit follows from his main statement:Theorem 4.2. Let 1 < r = limn→+∞sup |an|1/n < +∞ and z0 ∈ γ(r) be aregular point for the sum f(z) of (1.4) in E(r). Then t0 = w(z0) ∈ C(0, r) is aregular point for the sum g(t) of (4.1).Using Theorems 4.1 and 4.2 we are going to prove a theorem for theJacobi series, which is similar to Theorem 4.1.Theorem 4.3 [1]. Suppose that r from (1.4) satisfies the inequalities1 < r < +∞ and anrn = o(nδ) (n → +∞) for some δ > −1/2. Then the seriesEquiconvergence and equisummability of Jacobi series 329(1.4) is uniformly (C, δ − 1/2)-summable on every close arc γ1 ⊂ γ(r), all pointsof which are regular for the sum f(z) of (1.4).P r o o f. Let γ∗1 = {t : t = w(z), z ∈ γ1}. Obviously, γ∗1 is a close arc of thecircumference C(0, r). According to Theorem 4.2 all point of this arc are regularfor the sum g(t) of (4.1). Then, using Theorem 4.1 the series (4.1), we obtainthat this series is uniformly (C, δ)-summable on the arc γ∗1 . Then, the series+∞∑n=1antnis also uniformly (C, δ)-summable on the arc γ∗1 . Using [2, Theorem 76], we obtainthat the series+∞∑n=1ann−1/2tnis uniformly (C, δ − 1/2)-summable on the arc γ∗1 . Hence, the series (3.1) isuniformly (C, δ − 1/2)-summable for z ∈ γ1. According Theorem 3.2 the series(1.5) is uniformly (C, δ − 1/2)-summable on the arc γ1. Thus Theorem 4.3 isproved. Remark. The proof of Theorem 4.3 is considerably shorter than itsproof given in [1].REFERENCES[1] G. Boychev. Fatou-Riesz’s theorem for series in Jacobi polynomials. C. R.Acad. Bulgare Sci. 37, 4 (1984), 427–428.[2] G. Hardy. Divergent Series. Oxford, At the Clarendon Press (GeoffreyCumberlege), 1949.[3] P. Rusev. Classical Orthogonal Polynomials and Their Associated Func-tions in Complex Plane. Prof. Marin Drinov Academic Publushing House,Sofia, 2005.[4] G. Szego¨. Orthogonal Polynomials. AMS Colloquium Publ. vol. 23, NewYork, American Mathematical Society, 1939.330 Georgi Boychev[5] V. Yatsun. Analytic continuation of expansions in Jacobi polynomials.Ukra¨ın. Math. Zh. 21, 4 (1969), 511–521 (in Russian); English translationin Ukr. Math. J. 21 (1969), 436–444.[6] A. Zygmund. Trigonometric series. Volumes I and II, 3rd ed. CambridgeMathematical Library. Cambridge, Cambridge University Press, 2002.Trakia University6000 Stara Zagora, Bulgariae-mail: GBoychev@hotmail.com Received February 24, 2012

Equiconvergence and equisummability of Jacobi series

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Equiconvergence and equisummability of Jacobi series

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