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Using sums of squares to prove that certain

arXiv:math/9307210v1 [math.CA] 9 Jul 1993Using sums of squares to prove that certainentire functions have only real zerosbyGeorge Gasper1Dedicated to the memory of Ralph P. Boas, Jr. (1912–1992)(May 17, 1993 version)AbstractIt is shown how sums of squares of real valued functions can beused to give new proofs of the reality of the zeros of the Bessel func-tions Jα(z) when α ≥−1, confluent hypergeometric functions 0F1(c; z)when c > 0 or 0 > c > −1, Laguerre polynomials Lαn(z) when α ≥−2,and Jacobi polynomials P (α,β)n(z) when α ≥−1 and β ≥−1. Besidesyielding new inequalities for |F(z)|2, where F(z) is one of these func-tions, the derived identities lead to inequalities for ∂|F(z)|2/∂y and∂2|F(z)|2/∂y2, which also give new proofs of the reality of the zeros.1IntroductionIn a 1975 survey paper [11] on positivity and special functions it was shownhow sums of squares of special functions could be used to prove the nonneg-ativity of the Fej´er kernel, the positivity of integrals of Bessel functions [10]and of the Cotes’ numbers for some Jacobi abscissas, a Tur´an type inequalityfor Bessel functions, the Askey-Gasper inequality (cf.

[1], [2], [13], [15])nXk=0P (α,0)k(x) ≥0,α > −2,−1 ≤x ≤1,(1.1)1Supported in part by the National Science Foundation under grant DMS-9103177.Key words. Entire functions, inequalities, real zeros, sums of squares, confluent hyper-geometric functions, Bessel functions, Jacobi polynomials, Laguerre polynomials.1

which de Branges [5] employed to complete his proof of the Bieberbach con-jecture, and to prove the more general inequalities [12]nXk=0(λ + 1)kk! (λ + 1)n−k(n −k)!P (α,β)k(x)P (β,α)k(1)≥0,−1 ≤x ≤1,(1.2)when 0 ≤λ ≤α + β and β ≥−1/2.

It was also pointed out in [11] that,since one of Jensen’s necessary and sufficient conditions for the RiemannHypothesis to hold (given in P´olya [19]) is the condition thatZ ∞−∞Z ∞−∞Φ(s)Φ(t)ei(s+t)x(s −t)2ndsdt ≥0,−∞< x < ∞,(1.3)for n = 0, 1, 2, 3, . .

., whereΦ(t) = 2∞Xk=1(2k4π2e9t/2 −3k2πe5t/2)e−k2πe2t,(1.4)and the above integral is a square when n = 0, the method of sums of squaresis suggested for proving (1.3).Another of Jensen’s necessary and sufficient conditions for the RiemannHypothesis to hold is thatZ ∞−∞Z ∞−∞Φ(s)Φ(t)ei(s+t)xe(s−t)y(s −t)2dsdt ≥0,−∞< x, y < ∞,(1.5)which can also be written in the equivalent form∂2∂y2|Ξ(x + iy)|2 ≥0,−∞< x, y < ∞,(1.6)withΞ(z) =Z ∞−∞Φ(t) exp(izt)dt = 2Z ∞0Φ(t) cos(zt)dt. (1.7)That (1.6) is a sufficient condition for the Riemann Ξ(z) function to have onlyreal zeros follows directly from observation that, since |Ξ(x + iy)|2 = Ξ(x +iy)Ξ(x−iy) is a nonnegative even function of y, (1.6) implies that |Ξ(x+iy)|22

is a nonnegative even convex function of y with its unique minimum at y = 0,and hence Ξ(x + iy) ̸= 0 whenever y ̸= 0. If the function Φ(t) in (1.3) and(1.5) is replaced by a function Ψ(t) such that the conditions stated in [19,§1], are satisfied, then, by [19, pp.

17, 18], the inequalities in (1.3) and (1.5)are necessary and sufficient conditions for the Fourier (or cosine) transformof Ψ(t) to have only real zeros. In 1913 Jensen [18] proved that each of theinequalitiesy ∂∂y|F(x + iy)|2 ≥0,∂2∂y2|F(x + iy)|2 ≥0,−∞< x, y < ∞,(1.8)is necessary and sufficient for a real entire function F(z) ̸≡0 of genus 0 or 1(cf.

Boas [4, Chapter 2]) to have only real zeros. Also see Titchmarsh [21]and Varga [22, Chapter 3].In view of these observations and the successes of the sums of squaresmethod (also see [14], [16, Chapter 8]), since the early 1970’s I have beeninvestigating how squares of real valued functions can be used to prove thatcertain entire functions have only real zeros and to prove inequalities of theform in (1.8).In this paper I demonstrate how certain series expansionsin sums of squares of special functions give new proofs of the reality of thezeros of the Bessel functions Jα(z) when α ≥−1, confluent hypergeometricfunctions 0F1(c; z) when c > 0 or 0 > c > −1, Laguerre polynomials Lαn(z)when α ≥−2, and Jacobi polynomials P (α,β)n(z) when α ≥−1 and β ≥−1.Here, as elsewhere, z = x + iy is a complex variable and x and y are realvariables.For the definitions of these functions and their properties, seeErd´elyi [9] and Szeg˝o [20].

In addition, it will be shown that besides yieldingnew inequalities for |F(z)|2, where F(z) is one of these functions, the derivedidentities lead to inequalities for ∂|F(z)|2/∂y and ∂2|F(z)|2/∂y2, which alsogive new proofs of the reality of the zeros.2Initial observationsIn order to see how easily sums of squares can be used to prove that all ofthe zeros of sin z and cos z are real, it suffices to observe that we have the(easily verified) identities| sin z|2 = sin2 x + sinh2 y,(2.1)3

| cos z|2 = cos2 x + sinh2 y(2.2)and to note that sinh y = (ey −e−y)/2 > 0 when y > 0, and sinh y < 0 wheny < 0.One can also take partial derivatives of the identities in (2.1) and (2.2)with respect to y to obtain∂∂y| sin z|2 = ∂∂y| cos z|2 = sinh 2y(2.3)which shows that | sin z|2 and | cos z|2 are increasing (decreasing) functionsof y when y > 0 (y < 0), and to obtain∂2∂y2| sin z|2 = ∂2∂y2| cos z|2 = 2 cosh 2y = 2(cosh2 y + sinh2 y) ≥2,(2.4)which shows that | sin z|2 and | cos z|2 are convex functions of y. Then, be-cause | sin z|2 and | cos z|2 are nonnegative even functions of y, it immediatelyfollows from (2.3) and (2.4) that sin z and cos z have only real zeros.Observe that the reality of the zeros of sin z and cos z also follows fromthe inequalities| sin z|2 > sin2 x,| cos z|2 > cos2 x,(y ̸= 0)(2.5)| sin z|2 ≥sinh2 y,| cos z|2 ≥sinh2 y,(2.6)y ∂∂y| sin z|2 = y ∂∂y| cos z|2 ≥2y2,(2.7)∂2∂y2| sin z|2 = ∂2∂y2| cos z|2 ≥2 cosh2 y(2.8)∂2∂y2| sin z|2 = ∂2∂y2| cos z|2 ≥2 + 2 sinh2 y(2.9)which are consequences of (2.1)–(2.4).4

3Bessel functions and 0F1(c; z) functionsSince the identities and inequalities in §2 give the reality of the zeros of theBessel functions [20, (1.71.2)]J−12(z) = ( 2πz)12 cos z,J 12(z) = ( 2πz)12 sin z,(3.1)this suggests that it should be possible to use sums of squares to proveLommel’s theorem (see Watson [23, p. 482]) that all of the zeros of theBessel function [9, 7.2(3)]Jα(z) =(z/2)αΓ(α + 1) 0F1(α + 1; −z2/4)(3.2)are real when α > −1. With this aim in mind and in order to work withentire functions, we setJα(z) = z−αJα(z) =2−αΓ(α + 1)0F1(α + 1; −z2/4),(3.3)which is an even entire function of z such that Jα(z) = Jα(z) when α is real.Let α > −1.

Then, from the product formula (37) in Carlitz [8],|Jα(z)|2 =∞Xk=0(α + 12)k2k−αk! (2α + 1)kΓ(α + 1)(x2 + y2)kJα+k(2x).

(3.4)To express each of the Bessel functions on the right side of (3.4) as a sumof squares of real valued Bessel functions observe that from the additiontheorem for Bessel functions [9, 7.15(30)] we have the expansionJα+k(2x) = 2k+αΓ(k + α)∞Xj=0(j + k + α)(2k + 2α)jj!(−1)jx2j(Jα+j+k(x))2. (3.5)Hence, substituting (3.5) into (3.4) and changing the order of summation wefind that(3.6)|Jα(z)|2 =∞Xn=0(n + α)(2α)nα n!

(−1)nx2n× 2F1(−n, n + 2α; 2α + 1; 1 + y2/x2)(Jα+n(x))2.5

Now apply the Euler transformation formula [9, 2.9(3)]2F1(a, b; c; z) = (1 −z)−a2F1(a, c −b; c; z/(z −1))(3.7)to the above 2F1 series to obtain the desired sum of squares expansion formula|Jα(z)|2 = (Jα(x))2 + 2(α + 1)y2(Jα+1(x))2(3.8)+∞Xn=2(2n + 2α)(2α + 1)n−1n!y2n× 2F1(−n, 1 −n; 2α + 1; 1 + x2/y2)(Jα+n(x))2.When n ≥2, α > −1 and y ̸= 0, the positivity of the coefficients of(Jα+n(x))2 in (3.8) follows from(3.9)(2α + 1) 2F1(−n, 1 −n; 2α + 1; 1 + x2/y2) = (2α + 1 + n2 −n)+ n(n −1)x2/y2 +nXk=2(−n)k(1 −n)kk! (2α + 2)k−1(1 + x2/y2)k > 0.Hence, since the real zeros of Jα(x) and Jα+1(x) are interlaced, (3.8) givesa sum of squares proof that the entire functions Jα(z), and thus the Besselfunctions Jα(z), have only real zeros when α > −1.Letting α →−1 itfollows that the Bessel function J−1(z) = limα→−1 Jα(z) = −J1(z) has onlyreal zeros.Notice that the inequality|Jα(z)|2 ≥(Jα(x))2 + 2(α + 1)(yJα+1(x))2 > 0, y ̸= 0, α > −1,(3.10)and in fact infinitely many inequalities follow from (3.8) by just droppingterms from the right side of (3.8).

Analogous to (2.7)–(2.9), it follows bydifferentiating equation (3.6) with respect to y and applying (3.7) that wealso have the identities(3.11)y ∂∂y|Jα(z)|2 = 4y2∞Xn=0(n + α + 1)(2α + 2)nn!y2n× 2F1(−n, −n; 2α + 2; 1 + x2/y2)(Jα+n+1(x))26

and∂2∂y2|Jα(z)|2=4∞Xn=0(n + α + 1)(2α + 2)nn!y2n(3.12)×2F1(−n, −n; 2α + 2; 1 + x2/y2)(Jα+n+1(x))2+8y2∞Xn=0(n + α + 2)(2α + 3)n+1n!y2n×2F1(−n, −n −1; 2α + 3; 1 + x2/y2)(Jα+n+2(x))2,which give infinitely many inequalities, such as, e.g.,y ∂∂y|Jα(z)|2 ≥4(α + 1)(yJα+1(x))2 ≥0,α ≥−1,(3.13)∂2∂y2|Jα(z)|2 ≥4(α + 1)(Jα+1(x))2 ≥0,α ≥−1,(3.14)each of which proves that Jα(z) has only real zeros when α ≥−1.In view of (3.3) the reality of the zeros of Jα(z) when α > −1 is equivalentto the statement that all of the zeros of the confluent hypergeometric function0F1(c; z) are real and negative when c > 0. However, it is known [17] thatthe zeros of 0F1(c; z) are also real (but not necessarily negative) when −1

Because this fact does not follow from (3.8) or (3.11)–(3.14), we willnext show how it can also be proved by using sums of squares of real valuefunctions.From formulas (53) and (52) in Burchnall and Chaundy [7] it follows thatif c is real valued and c ̸= 0, −1, −2, . .

. , then we have the expansion formulas|0F1(c; z)|2 =∞Xk=01k!

(c)k(c)2k(x2 + y2)k0F1(c + 2k; 2x)(3.15)and0F1(c + 2k; 2x) =∞Xj=0(−1)jj! (c + 2k + j −1)j(c + 2k)2jx2j (0F1(c + 2k + 2j; x))2 .

(3.16)7

As in the Bessel function case, substitute (3.16) into (3.15) and change theorder of summation to get(3.17)|0F1(c; z)|2 =∞Xn=0(−1)nn! (c + n −1)n(c)2nx2n× 2F1(−n, n + c −1; c; 1 + y2/x2)(0F1(c + 2n; x))2which, by applying the transformation formula (3.7), gives(3.18)|0F1(c; z)|2 =∞Xn=01n!

(n + c −1)n(c)2ny2n× 2F1(−n, 1 −n ; c; 1 + x2/y2)(0F1(c + 2n; x))2.When c > 0 and y ̸= 0 the coefficient of (0F1(c + 2n; x))2 in the series in(3.18) is obviously positive. Hence, since 0F1(c; x) > 0 when c > 0 and x ≥0,(3.18) gives another proof that 0F1(c; z) has only real negative zeros whenc > 0.To handle the case −1 < c < 0 differentiate equation (3.17) with respectto y and apply (3.7) to obtain(3.19)y ∂∂y |c 0F1(c; z)|2 = 2y2∞Xn=0(c + 1)nn!

(c + 1)2n(c + 1)2n+1y2n× 2F1(−n, −n; c + 1; 1 + x2/y2)(0F1(c + 2n + 2; x))2and∂2∂y2 |c 0F1(c; z)|2 = 2∞Xn=0(c + 1)nn! (c + 1)2n(c + 1)2n+1y2n(3.20)× 2F1(−n, −n; c + 1; 1 + x2/y2)(0F1(c + 2n + 2; x))2+ 4y2∞Xn=0(c + 2)n+1n!

(c + 1)2n+2(c + 1)2n+3y2n× 2F1(−n, −n −1; c + 2; 1 + x2/y2)(0F1(c + 2n + 4; x))2which, in particular, give the inequalitiesy ∂∂yc(c + 1) 0F1(c; z)2 ≥2(c + 1)(y 0F1(c + 2; x))2,c ≥−1,(3.21)8

and∂2∂y2 |c(c + 1) 0F1(c; z)|2 ≥2(c + 1)(0F1(c + 2; x))2,c ≥−1. (3.22)Since the coefficients on the right hand sides of (3.19)–(3.22) are clearlypositive when c > −1 and y ̸= 0, these formulas prove that the functionsc(c + 1) 0F1(c; z) have only real zeros when c ≥−1, where it is understoodthat c(c + 1) 0F1(c; z) is to be replaced by its c →0 limit case z 0F1(2; z)when c = 0, and by its c →−1 limit case z2 0F1(3; z)/2 when c = −1.4Laguerre polynomials and 1F1(a; c; z) func-tionsWhen α > −1 the Laguerre polynomialsLαn(z) = (α + 1)nn!1F1(−n; α + 1; z)(4.1)satisfy the orthogonality relationZ ∞0Lαn(x)Lαm(x)xαe−x dx = Γ(n + α + 1)n!δnm, n, m = 0, 1, 2, .

. .

,(4.2)from which it follows by a standard argument (cf. [20, §3.3]) that the zerosof Lαn(z) are real and positive.

Analogous to the last part of the previoussection, in this section we will derive some sums of squares expansions which,besides proving the reality of the zeros of these polynomials when α > −1,also prove that they have only real zeros (not necessarily positive) when−1 ≥α ≥−2, where Lαn(z) is defined to be the α →−k limit case of (4.1)when α is a negative integer −k. Thus Lα1(z) = α+1−z, which has a negativezero when α < −1, and Lα2(z) = ((α + 1)(α + 2) −2(α + 2)z + z2).2, whichhas non-real zeros when α < −2.9

Let α be real valued. Substituting the sum of squares of Laguerre poly-nomials expansion (from [7, (91)])Lα+2kn−k (2x) =n−kXj=0(n −k −j)!

(2k + 2j + α)(2k + α)jj! (2k + α)(2k + α + 1)n+j−k(4.3)× (−1)jx2j Lα+2k+2jn−k−j(x)2into the special case of [3, (5.4)]|Lαn(z)|2 = (α + 1)nn!nXk=01k!

(α + 1)k(x2 + y2)kLα+2kn−k (2x)(4.4)and changing the order of summation yields(4.5)|Lαn(z)|2 = (α + 1)nn!nXk=0(n −k)! (2k + α)(α)kk!

α(α + 1)n+k(−1)kx2k× 2F1(−k, k + α; α + 1; 1 + y2/x2)Lα+2kn−k (x)2 .Then application of (3.7) gives(4.6)|Lαn(z)|2 = (α + 1)nn!nXk=0(n −k)! (2k + α)(α)kk!

α(α + 1)n+ky2k× 2F1(−k, 1 −k; α + 1; 1 + x2/y2)Lα+2kn−k (x)2 .Since Lα0(x) ≡1 and the coefficients on the right hand side of (4.6) areclearly positive when α > −1 and y ̸= 0, the expansion (4.6) proves that theLaguerre polynomials have only real zeros when α > −1. This also follows,in particular, from the inequalities|Lαn(z)|2 ≥(α + 1)nn!

n! (n + α)ny2n2F1(−n, 1 −n; α + 1; 1 + x2/y2),α > −1,(4.7)and|Lαn(z)|2 ≥|Lαn(x)|2 +(α + 1)nn!

n! (n + α)ny2n,α > −1, n ≥1,(4.8)10

which are consequences of (4.6).Now differentiate equation (4.5) with respect to y and apply (3.7) toderive the expansions(4.9)y ∂∂y |Lαn(z)|2 = 2y2n−1Xk=0(n −k −1)! (2k + α + 2)(α + 2)kn!

k! (n + α + 1)k+1y2k× 2F1(−k, −k; α + 2; 1 + x2/y2)Lα+2k+2n−k−1 (x)2 ,n ≥1,and∂2∂y2 |Lαn(z)|2 = 2n−1Xk=0(n −k −1)!

(2k + α + 2)(α + 2)kn! k!

(n + α + 1)k+1y2k(4.10)× 2F1(−k, −k; α + 2; 1 + x2/y2)Lα+2k+2n−k−1 (x)2+ 4y2n−2Xk=0(n −k −2)! (2k + α + 4)(α + 3)k+1n!

k! (n + α + 1)k+2y2k× 2F1(−k, −k −1; α + 3; 1 + x2/y2)Lα+2k+4n−k−2 (x)2,n ≥1,which yield, e.g., the inequalitiesy ∂∂y |Lαn(z)|2 ≥2(α + 2)n(n + α + 1)yLα+2n−1(x)2 ,α > −2, n ≥1,(4.11)and∂2∂y2 |Lαn(z)|2 ≥2(α + 2)n(n + α + 1)Lα+2n−1(x)2 ,α > −2, n ≥1,(4.12)and prove (after letting α →−2) that the polynomials Lαn(z) have only realzeros when α ≥−2.For the confluent hypergeometric functions 1F1(a; c; z) with a and c realvalued and c ̸= 0, −1, −2, .

. .

, use of the expansion formulas [7, (42) and(43)] instead of (4.3) and (4.4) yields the nonterminating extension of (4.5)(4.13)|1F1(a; c; z)|2 =∞Xk=0(a)k(c −a)kk! (c)2k(c + k −1)kx2k× 2F1(−k, c + k −1; c; 1 + y2/x2)(1F1(a + k; c + 2k; x))211

and hence, by (3.7),(4.14)|1F1(a; c; z)|2 =∞Xk=0(a)k(c −a)kk! (c)2k(c + k −1)k(−1)ky2k× 2F1(−k, 1 −k; c; 1 + x2/y2)(1F1(a + k; c + 2k; x))2.Then differentiation of equation (4.13) with respect of y and application of(3.7) gives the following extensions of (4.9) and (4.10) (and also of (3.19) and(3.20)), respectively,(4.15)y ∂∂y|c(c + 1) 1F1(a; c; z)|2 = 2y2∞Xk=0(a)k+1(c −a)k+1(c + 1)k!

(c + 2)2k(c + k + 1)k(−1)k+1y2k× 2F1(−k, −k; c + 1; 1 + x2/y2)(1F1(a + k + 1; c + 2k + 2; x))2and∂2∂y2 |c(c + 1) 1F1(a; c; z)|2 = 2∞Xk=0(a)k+1(c −a)k+1(c + 1)k! (c + 2)2k(c + k + 1)k(−1)k+1y2k(4.16)× 2F1(−k, −k; c + 1; 1 + x2/y2)(1F1(a + k + 1; c + 2k + 2; x))2+ 4y2∞Xk=0(a)k+2(c −a)k+2k!

(c + 2)2k+2(c + k + 3)k(−1)ky2k× 2F1(−k, −k −1; c + 2; 1 + x2/y2)(1F1(a + k + 2; c + 2k + 4; x))2.If a = −n is a negative integer and c = α + 1, then (4.13)–(4.16) reduceto (4.5), (4.6), (4.9), (4.10), respectively. If a = c + n with n a nonnegativeinteger, then (4.15) and (4.16) reduce to terminating sums of squares expan-sions with nonnegative coefficients which prove that c(c + 1) 1F1(c + n; c; z),as a function of z, has only real zeros when c ≥−1, where this function isto be replaced by its c →0 and c →−1 limit cases when c = 0 and c = −1,respectively.

It should be noted that, in view of Kummer’s transformationformula [9, 6.3(7)]1F1(a; c; x) = ex1F1(c −a; c; −x),(4.17)these results on the zeros of c(c + 1) 1F1(c + n; c; z) are equivalent to thoseobtained above for the Laguerre polynomials.12

5Jacobi polynomialsWhen α > −1 and β > −1 the Jacobi polynomialsP (α,β)n(z) = (α + 1)nn!2F1(−n, n + α + β + 1; α + 1; (1 −z)/2)(5.1)satisfy the orthogonality relationZ 1−1 P (α,β)n(x)P (α,β)m(x)(1 −x)α(1 + x)β dx = 0,n ̸= m,(5.2)for n, m = 0, 1, 2, . .

. , and hence, by [20, Theorem 3.3.1], these polynomialshave only real zeros.

In our derivation of sums of squares expansions whichimply the reality of the zeros of these polynomials we will start out by derivingsums of squares expansions for nonterminating 2F1(a, b; c; z) hypergeometricseries with |z| < 1 (for convergence).Let a, b, c be real valued, c ̸= 0, −1, −2, . .

. , and |z| < 1.

Then formula [6,(51)] gives the expansion|2F1(a, b; c; z)|2 =∞Xk=0(a)k(b)k(c −a)k(c −b)kk! (c)k(c)2k(x2 + y2)k(5.3)× 2F1(a + k, b + k; c + 2k; 2x −x2 −y2).Unfortunately, application of the inversion [6, (50)] of [6, (51)] to each of the2F1(a + k, b + k; c + 2k; 2x −x2 −y2) functions on the right side of equation(5.3) just returns one back to the function that is on the left side.

Therefore,we use formulas (44), (45), (50) in [6] to obtain, respectively, the expansions(5.4)2F1(a + k, b + k; c + 2k; 2x −x2 −y2)=∞Xj=0(a + k)j(b + k)jj! (c + 2k)j(−1)j(x2 +y2)j2F1(a+k +j, b+k +j; c+2k +j; 2x),(5.5)2F1(a + k + j, b + k + j; c + 2k + j; 2x) =∞Xm=0(a + k + j)m(b + k + j)mm!

(c + 2k + j)mx2m× 2F1(a + k + j + m, b + k + j + m; c + 2k + j + m; 2x −x2),13

(5.6)2F1(a + k + j + m, b + k + j + m; c + 2k + j + m; 2x −x2)=∞Xn=0(a + k + j + m)n(b + k + j + m)n(c −a + k)n(c −b + k)nn! (c + 2k + j + m + n −1)n(c + 2k + j + m)2n(−1)nx2n× (2F1(a + k + j + m + n, b + k + j + m + n; c + 2k + j + m + 2n; x))2,and then substitute these expansions in turn into (5.3), change the order ofsummation and use the binomial theorem to obtain(5.7)|2F1(a, b; c; z)|2 =∞Xm=0mXj=0(a)m(b)m(c −a)j(c −b)jj!

(m −j)! (c)m+j(m + c −1)j(−1)mx2jy2m−2j× 2F1(−j, m + c −1; c; 1 + y2/x2)(2F1(m + a, m + b; m + j + c; x))2.Application of (3.7) to the first 2F1 series on the right side of (5.7) gives(5.8)|2F1(a, b; c; z)|2 =∞Xm=0mXj=0(a)m(b)m(c −a)j(c −b)jj!

(m −j)! (c)m+j(m + c −1)j(−1)m+jy2m× 2F1(−j, 1 −m; c; 1 + x2/y2)(2F1(m + a, m + b; m + j + c; x))2,which contains (4.14) as a limit case.

When a = −n is a negative integer,b = n + α + β + 1 and c = α + 1, it follows from (5.8) that(5.9)n! (α + 1)nP (α,β)n(1 −2z)2=nXm=0mXj=0(−n)m(n + α + β + 1)m(n + α + 1)j(−n −β)jj!

(m −j)! (α + 1)m+j(m + α)j(−1)m+jy2m×2F1(−j, 1−m; α+1; 1+x2/y2)(2F1(m−n, m+n+α+β+1; m+j+α+1; x))2,which gives a sums of squares proof that the Jacobi polynomials P (α,β)n(z)have only real zeros when α, β > −1 (since the coefficients in (5.9) are thenclearly positive) and hence, by continuity, when α, β ≥−1.

The restrictionthat α, β ≥−1 cannot be extended to α, β ≥−2 because P (α,β)2(z) hasnon-real zeros when α, β > −2 and α + β < −3.As in sections 3 and 4 one may repeatedly differentiate (5.7) with respectto y and apply (3.7) to obtain extensions of (4.15), (4.16), etc. But, since theresulting identities are quite lengthly and do not add any additional (α, β)14

for which the Jacobi polynomials have only real zeros, we will omit them andonly point out that the first two differentiations give identities that yield, inparticular, the inequalitiesy ∂∂yP (α,β)n(1 −2z)2 ≥2n(n + α + β + 1)n(α + 1)nn! n!y2n(5.10)and∂2∂y2P (α,β)n(1 −2z)2 ≥2n(2n −1)(n + α + β + 1)n(α + 1)nn!

n!y2n−2(5.11)when n ≥1 and α, β ≥−1.In subsequent papers it will be shown that squares of real valued functionscan also be used to prove the reality of the zeros of some non-classical familiesof orthogonal polynomials, of the cosine transformsZ ∞0e−a cosh t cos zt dt,a > 0,and of some other entire functions.References[1] R. Askey and G. Gasper, Positive Jacobi polynomial sums II, Amer. J.

Math. 98(1976), 709–737.

[2] R. Askey and G. Gasper, Inequalities for polynomials, in The Bieberbach Conjecture,Proceedings of the Symposium on the Occasion of the Proof, Surveys and Monographs,No. 21, Amer.

Math. Soc., Providence, RI (1986), 7–32.

[3] W.N. Bailey, On the product of two Legendre polynomials with different arguments,Proc.

London Math. Soc.

(2) 41 (1936), 215–220. [4] R.P.

Boas, Entire Functions, Academic Press, Inc., New York, 1954. [5] L. de Branges, A proof of the Bieberbach conjecture, Acta Math.

154 (1985), 137–152. [6] J.L.

Burchnall and T.W. Chaundy, Expansions of Appell’s double hypergeometricfunctions, Quart.

J. Math.

(Oxford) 11 (1940), 249–270. [7] J.L.

Burchnall and T.W. Chaundy, Expansions of Appell’s double hypergeometricfunctions (II), Quart.

J. Math.

(Oxford) 12 (1941), 112–128.15

[8] L. Carlitz, Some polynomials related to the ultraspherical polynomials, PortugaliaeMath. 20 (1961), 127–136.

[9] A. Erdelyi, Higher Transcendental Functions, vols. 1 and 2, McGraw Hill, New York,1953.

[10] G. Gasper, Positive integrals of Bessel functions, SIAM J. Math.

Anal. 6 (1975),868–881.

[11] G. Gasper, Positivity and special functions, in Theory and Applications of SpecialFunctions, R. Askey, ed., Academic Press, New York (1975), 375–433. [12] G. Gasper, Positive sums of the classical orthogonal polynomials, SIAM J. Math.Anal.

8 (1977), 423–447. [13] G. Gasper, A short proof of an inequality used by de Branges in his proof of theBieberbach, Robertson and Milin conjectures, Complex Variables: Theory Appl.

7(1986), 45–50. [14] G. Gasper, q-Extensions of Clausen’s formula and of the inequalities used by deBranges in his proof of the Bieberbach, Robertson, and Milin conjectures, SIAM J.Math.

Anal. 20 (1989), 1019–1034.

[15] G. Gasper, Using symbolic computer algebraic systems to derive formulas involv-ing orthogonal polynomials and other special functions, in Orthogonal Polynomials:Theory and Practice, ed. by P. Nevai, Kluwer Academic Publishers, Boston, 1989,163–179.

[16] G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge UniversityPress, 1990. [17] E. Hille, Note on some hypergeometric series of higher order, J. London Math.

Soc.4 (1929), 50–54. [18] J.L.W.V.

Jensen, Recherches sur la th´eorie des ´equations, Acta Math. 36 (1913),181–195.

[19] G.P´olya,¨Uberdiealgebraisch-funktionentheoretischenUntersuchungenvonJ. L. W. V. Jensen, Kgl.

Danske Videnskabernes Selskab. Math.-Fys.

Medd. 7 (17)(1927), pp.

3–33; reprinted in his Collected Papers, Vol. II, pp.

278–308. [20] G. Szeg˝o, Orthogonal Polynomials, 4th ed., Amer.

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Colloq. Publ.

23, Prov-idence, R.I. 1975.

[21] E.C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd edition (Revisedby D.R.

Heath-Brown), Oxford Univ. Press, Oxford and New York, 1986.

[22] R.S. Varga, Scientific Computation on Mathematical Problems and Conjectures,CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadel-phia, 1990.

[23] G.N. Watson, Theory of Bessel Functions, Cambridge Univ.

Press, Cambridge andNew York, 1944.16

George GasperDepartment of MathematicsNorthwestern UniversityEvanston, IL 60208E-Mail: g-gasper@nwu.edu17

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