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  • Journal of Global Optimization manuscript No. (will be inserted by the editor)

    Some properties of a hypergeometric function which appear in an approximation problem

    Gradimir V. Milovanović · Michael Th. Rassias

    Received: date / Accepted: date

    Abstract In this paper we consider properties and power expressions of the functions f : (−1,1)→ R and fL : (−1,1)→ R, defined by

    f (x;γ) = 1 π

    ∫ 1 −1

    (1+ xt)γ√ 1− t2

    dt and fL(x;γ) = 1 π

    ∫ 1 −1

    (1+ xt)γ log(1+ xt)√ 1− t2

    dt,

    respectively, where γ is a real parameter, as well as some properties of a two para- metric real-valued function D( · ;α,β ) : (−1,1)→ R, defined by

    D(x;α,β ) = f (x;β ) f (x;−α−1)− f (x;−α) f (x;β −1), α,β ∈ R.

    The inequality of Turán type

    D(x;α,β )> 0, −1 < x < 1,

    for α +β > 0 is proved, as well as an opposite inequality if α +β < 0. Finally, for the partial derivatives of D(x;α,β ) with respect to α or β , respectively A(x;α,β ) and B(x;α,β ), for which A(x;α,β ) = B(x;−β ,−α), some results are obtained.

    We mention also that some results of this paper have been successfully applied in various problems in the theory of polynomial approximation and some “truncated”

    The first author was supported in part by the Serbian Ministry of Education, Science and Technological Development.

    G. V. Milovanović Mathematical Institute Serbian Academy of Science and Arts Kneza Mihaila 36, 11001 Belgrade, Serbia E-mail: gvm@mi.sanu.ac.rs

    M. Th. Rassias Department of Mathematics ETH-Zentrum, Rämistrasse 101 CH–8092 Zurich, Switzerland E-mail: michail.rassias@math.ethz.ch

  • 2

    quadrature formulas of Gaussian type with an exponential weight on the real semi- axis, especially in a computation of Mhaskar–Rahmanov–Saff numbers.

    Keywords Approximation · Expansion · Minimum · Maximum · Turán type inequality · Hypergeometric function · Gamma function · Digamma function

    Mathematics Subject Classification (2000) 26D07 · 26D15 · 33C05 · 41A10 · 41A17 · 49K05

    1 Introduction

    In a computation of Mhaskar–Rahmanov–Saff numbers in some problems of poly- nomial approximation on (0,+∞) and the corresponding “truncated” quadrature for- mulas of Gaussian type with respect to the weight function w(x) = exp(−x−α − xβ ), α > 0, β ≥ 1, on the real semiaxis, a one-parametric function f : (−1,1)→R defined by

    f (x;γ) = 1 π

    ∫ 1 −1

    (1+ xt)γ√ 1− t2

    dt, γ ∈ R, (1)

    is introduced. In this article we study some properties and power expansions of the function (1),

    as well as of the function fL : (−1,1)→ R, defined by

    fL(x;γ) = 1 π

    ∫ 1 −1

    (1+ xt)γ log(1+ xt)√ 1− t2

    dt, γ ∈ R. (2)

    We prove certain inequalities of Turán type including the function (1) and two real parameters. In 1941 the well-known Hungarian mathematician Paul Turán discovered the following inequality

    ∆n(x) = Pn(x)2−Pn−1(x)Pn+1(x)> 0, −1 < x < 1; n ∈ N,

    where Pn are the classical Legendre polynomials. However, it was published nine years later [19] (see also Szegő [18]). There are several extensions and generaliza- tions of this nice and simple inequality in different ways for several classes of or- thogonal polynomials, special functions, etc. L. Alpár [1], who was a Ph.D student of Turán, mentioned that this inequality of Turán had a wide-ranging influence in a number of disciplines (see also a recent paper by Baricz, Jankov, and Pogány [9] that has just been published). Some of these results have been successfully applied in various problems which arise in information theory (cf. [15]), numerical analysis and approximation theory, economic theory, biophysics, etc. For some further recent results one can see the papers [3], [6], [11], [7], [8], [2], [5], [10], among others.

    The function (1) can be expressed in terms of the hypergeometric function

    2F1(a,b;c;z) = +∞

    ∑ k=0

    (a)k(b)k (c)k

    zk

    k!

    as follows

    f (x;γ) = 2F1 (

    1− γ 2

    ,−γ 2

    ;1;x2 ) . (3)

  • 3

    Here (a)k denotes Pochhammer’s symbol that is defined by

    (a)k = a(a+1) · · ·(a+ k−1) = Γ (a+ k)

    Γ (a) , (a)0 = 1,

    where Γ is the Euler gamma function. The function f (x;γ) is even, with f (0;γ) = 1, so that we can consider it only on

    [0,1). Thus, for 0 < x < 1 we consider the following determinant of Turán type

    D(x;α,β ) =

    ∣∣∣∣∣ f (x;β ) f (x;−α)f (x;β −1) f (x;−α−1) ∣∣∣∣∣ , (4)

    as well as its partial derivatives with respect to the parameters α and β , i.e.,

    A(x;α,β ) = ∂

    ∂α D(x;α,β ) and B(x;α,β ) =

    ∂ ∂β

    D(x;α,β ), (5)

    respectively. Therefore, we need the corresponding even function fL : (−1,1)→ R defined by (2).

    Remark 1 In the computation of Mhaskar–Rahmanov–Saff numbers (cf. [14]) for an exponential weight function on (0,+∞), the following nonlinear equation in x,

    β (

    α β · f (x;−α−1)

    f (x;β −1)

    ) β α+β

    f (x;β )−α (

    β α · f (x;β −1)

    f (x;−α−1)

    ) α α+β

    f (x;−α) = t,

    must be solved for α > 0, β ≥ 1, and a given positive t, where f is defined by (1), i.e., (3). The equation has a unique solution in (0,1).

    This paper is organized as follows. Some important properties of functions f (x;γ) and fL(x;γ) and their power expansions are given in Section 2. Section 3 is devoted to properties of the determinant D(x;α,β ), including an inequality of Turán type, as well as a power expansion of D(x;α,β ). Finally, in Section 4 some properties of A(x;α,β ) and B(x;α,β ) are presented.

    2 Some Properties of the Functions f (x;γ) and fL(x;γ)

    According to the representation (3), by trivially rewriting the Pochhammer symbols as binomial coefficients,( −γ

    2

    ) k

    ( −γ

    2 +

    1 2

    ) k = 2−2kγ(γ−1) · · ·(γ− (2k−1)) = 2−2k

    ( γ 2k

    )( 2k k

    ) (k!)2,

    we get the following power expansion of the function (1).

    Theorem 1 For |x|< 1 we have

    f (x;γ) = +∞

    ∑ n=0

    ( γ

    2n

    )( 2n n

    )( x 2

    )2n . (6)

  • 4

    Remark 2 Since

    f (x;γ) = 1+ γ(γ−1)

    4 x2 +

    γ(γ−1)(γ−2)(γ−3) 64

    x4 + · · · ,

    we have

    f (x;0) = f (x;1) = 1, f (x;2) = f (x;3) = 1+ γ(γ−1)

    4 x2, etc.

    Also, it is easy to see that

    f (0;γ) = 1, f ′(0;γ) = 0, f ′′(0;γ) = γ(γ−1)

    2 , etc.

    The corresponding expansion of the function x 7→ fL(x;γ) can be done in a similar way.

    Theorem 2 For |x|< 1 we have

    fL(x;γ) = +∞

    ∑ n=1

    ( 2n n

    ){ 2n ∑ k=1

    (−1)k−1

    k

    ( γ

    2n− k

    )}( x 2

    )2n . (7)

    Alternatively, it can be expressed in the form

    fL(x;γ) = +∞

    ∑ n=1

    ω ′n(γ) (n!)2

    ( x 2

    )2n , (8)

    where ωn(γ) = γ(γ−1) · · ·(γ−2n+1).

    Proof Using the binomial expansion and the expansion of log(1+ t) in t, we have the following potential series

    (1+ t)γ log(1+ t) = +∞

    ∑ n=1

    { n

    ∑ k=1

    (−1)k−1

    k

    ( γ

    n− k

    )} tn, |t|< 1,

    for each γ ∈ R. Then, substituting t by xt and integrating it over (−1,1) with respect to the Chebyshev weight of the first kind, we obtain

    fL(x;γ) = +∞

    ∑ n=1

    { n

    ∑ k=1

    (−1)k−1

    k

    ( γ

    n− k

    )} 1 π

    ∫ 1 −1

    xntn√ 1− t2

    dt,

    i.e., (7). Using the Saalschütz formula [17, p. 11]

    n−1

    ∑ k=0

    ( z k

    ) xn−k

    n− k =

    n

    ∑ k=1

    ( z

    n− k

    ) xk

    k =

    n

    ∑ k=1

    ( z− k n− k

    ) (x+1)k−1

    k ,

    for x :=−1, z := γ , and by substituting n by 2n, we get 2n

    ∑ k=1

    (−1)k−1

    k

    ( γ

    2n− k

    ) =

    2n

    ∑ k=1

    ( γ− k

    2n− k

    ) 1 k =

    γ 2n

    ( γ−1

    2n−1

    ) [ψ(−γ)−ψ(2n− γ)],

  • 5

    where ψ is the digamma function, i.e.,( 2n n

    ) 2n ∑ k=1

    (−1)k−1

    k

    ( γ

    2n− k

    ) =

    ( 2n n

    )( γ

    2n

    )2n−1 ∑ k=0

    1 γ− k

    = ω ′n(γ) (n!)2

    ,

    because of d dγ

    (logωn(γ)) = ω ′n(γ) ωn(γ)

    = 2n−1

    ∑ k=0

    1 γ− k

    .

    The expansion (8) can be also obtained by a formal differentiation of (6) with respect to γ . ut

    The expansion (7), i.e., (8), gives

    fL(x;γ) = (2γ−1) ( x

    2

    )2 +

    1 2 ( 2γ3−9γ2 +11γ−3

    )( x 2

    )4 +

    1 36 ( 6γ5−75γ4 +340γ3−675γ2 +548γ−120

    )( x 2

    )6 +

    1 144

    ( 2γ7−49γ6 +483γ5−2450γ4 +6769γ3−9849γ2 +6534γ−1260

    )( x 2

    )8 +

    1 14400

    ( 10γ9−405γ8 +6960γ7−66150γ6 +379638γ5−1346625γ4

    +2894720γ3−3518100γ2 +2053152γ−362880 )( x

    2

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