## A Problem of Pólya Concerning Polynomials Which Obey Descartes' Rule of Signs (1997)

Venue: | East J. Approx |

Citations: | 2 - 2 self |

### BibTeX

@ARTICLE{Dimitrov97aproblem,

author = {Dimitar K. Dimitrov},

title = {A Problem of Pólya Concerning Polynomials Which Obey Descartes' Rule of Signs},

journal = {East J. Approx},

year = {1997},

volume = {3},

pages = {241--250}

}

### OpenURL

### Abstract

We formulate and duscuss two open problems. The first one is due to P'olya. It states that the sequence of polynomials formed by a polynomial p with only real zeros and all its derivatives, obeys Descartes' rule of signs for any x, greater than the largest zero of p. The other problem is due to Karlin and states that certain Hankel determinants associated with an entire function in the Laguerre-Polya class do not change their signs. We prove that the statement of Karlin's problem is a consequence of that of P'olya's problem. The interest in these problems is motivated by the fact that once Karlin's problem is solved, it would yield necessary conditions that the Riemann hypothesis holds. 1

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(Show Context)
Citation Context ...ompact sets of the complex plane of polynomials whose zeros are real and are all positive, or all negative. It is clear that L \Gamma P(I) ae L \Gamma P . An important observation of P'olya and Schur =-=[17]-=- is that, if a function '(x) = 1 X k=0 fl k x k k! (3) is in L \Gamma P and its Maclaurin coefficients fl k ; k = 0; 1; : : : ; are nonnegative, then ' 2 L \Gamma P(I), and, moreover, the constant ter... |

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Citation Context ...ive. 2 For any 2k \Gamma 2 times continuously differentiable function f , the k \Theta k Hankel determinant H k (f ; x) is defined by H k (f ; x) = det(f (i+j) (x)) k\Gamma1 i;j=0 : Problem 2 (Karlin =-=[8]-=-) Let ' 2 L \Gamma P and its Maclaurin coefficients fl k be nonnegative. Prove that for any q = 0; 1; : : : and k = 2; 3; : : : (\Gamma1) k(k\Gamma1)=2 H k (' (q) ; x)s0 for xs0: (4) As it was pointed... |

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Citation Context ... ); (2) where c is real, oes0; m is a nonnegative integer, x k ? 0, and P 1=x k ! 1. If ' is of type I in the Laguerre-P'olya class, we shall write ' 2 L \Gamma P(I). We have adopted the notations of =-=[4, 14, 18]-=- where one may consult for the properties of functions in the Laguerre-Polya class. Generally, L \Gamma P consists of entire functions which are uniform limits on the compact sets of the complex plane... |

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Citation Context ...ls, associated with ', converges locally uniformly to '. On the other hand, if ' is in L \Gamma P , then, for any positive integer n, the polynomial g n (z) has only real zeros [17] (see also [4] and =-=[6]-=-). Since the coefficients of g n (z) are nonnegative, then all its zeros are nonpositive. Thus, we may chose p n (z) = g n (z=n): (15) Since the convergence of p n (z) to '(z) is uniform on the compac... |

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Citation Context ...ents of F . Precisely, because of (11), P'olya's conjecture reads as (2q + 1)(sb q ) 2 \Gamma (2q \Gamma 1)sb q\Gamma1sb q+1 ? 0; q = 1; 2; : : : : Almost sixty years later Csordas, Norfolk and Varga =-=[5]-=- proved the latter inequalities, thus verifying P'olya's conjecture. We refer the reader to a nice account of Varga [20, Chapter 3] about the proof of the conjecture and some other theoretical and com... |

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Citation Context ...oof, we suppose that ' is not a polynomial and nsm. We consider two cases. The first one is when m = 0 in the representation (2) of ', and ' is not of the form ' = cx m e oex . Csordas and Williamson =-=[7]-=- proved that in this case the zeros of the corresponding Jensen polynomials are simple. Thus, the polynomial p n , defined by (15) has also simple negative zeros and positive leading coefficient. Let ... |

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Citation Context ... ); (2) where c is real, oes0; m is a nonnegative integer, x k ? 0, and P 1=x k ! 1. If ' is of type I in the Laguerre-P'olya class, we shall write ' 2 L \Gamma P(I). We have adopted the notations of =-=[4, 14, 18]-=- where one may consult for the properties of functions in the Laguerre-Polya class. Generally, L \Gamma P consists of entire functions which are uniform limits on the compact sets of the complex plane... |

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Citation Context ... with distinct real zeros j 1 ! : : : ! j n . Prove that the sequence p; p 0 ; : : : ; p (n) obeys Descartes' rule in (j n ; 1). This problem appears in a footnote of a celebrated paper of Schoenberg =-=[19]-=-. Schoenberg atributes it to P'olya. The statement of the problem is very natural. Note that the functions 1; x; : : : ; x n obey Descartes' rule in (0; 1) and the proof of this fact goes back to Desc... |

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Citation Context ...at the sequence of Legendre polynomials P 0 ; P 1 ; : : : ; P n obeys the Descartes' rule for x ? i n , where i n is the largest zero of P n (x) was proved much earlier by Laguerre [9], and M. Marden =-=[10]-=- gave a proof of Obrechkoff-Schoenberg's result for the classical orthogonal polynomials. Laguerre's and Marden's proofs use the second order differential equation satisfied by the classical orthogona... |

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Citation Context ...ferent. The fact that the sequence of Legendre polynomials P 0 ; P 1 ; : : : ; P n obeys the Descartes' rule for x ? i n , where i n is the largest zero of P n (x) was proved much earlier by Laguerre =-=[9]-=-, and M. Marden [10] gave a proof of Obrechkoff-Schoenberg's result for the classical orthogonal polynomials. Laguerre's and Marden's proofs use the second order differential equation satisfied by the... |

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Citation Context ...icients of all the polynomials have the same sign, also obeys Descartes' rule for x ? i n , where i n is the largest zero of p n (x). The latter result, as stated here, was proved first by Obrechkoff =-=[11, 12, 13]-=- and rediscovered later by Schoenberg [19]. It is worth mentioning that the proofs of Obrechkoff and of Schoenberg are different. The fact that the sequence of Legendre polynomials P 0 ; P 1 ; : : : ;... |

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Citation Context ...icients of all the polynomials have the same sign, also obeys Descartes' rule for x ? i n , where i n is the largest zero of p n (x). The latter result, as stated here, was proved first by Obrechkoff =-=[11, 12, 13]-=- and rediscovered later by Schoenberg [19]. It is worth mentioning that the proofs of Obrechkoff and of Schoenberg are different. The fact that the sequence of Legendre polynomials P 0 ; P 1 ; : : : ;... |

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Citation Context ...icients of all the polynomials have the same sign, also obeys Descartes' rule for x ? i n , where i n is the largest zero of p n (x). The latter result, as stated here, was proved first by Obrechkoff =-=[11, 12, 13]-=- and rediscovered later by Schoenberg [19]. It is worth mentioning that the proofs of Obrechkoff and of Schoenberg are different. The fact that the sequence of Legendre polynomials P 0 ; P 1 ; : : : ;... |

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Citation Context ...q+1sfl q+2 \Delta \Delta \Deltasfl q+k . . . . . . . . . . . .sfl q+k\Gamma1sfl q+k \Delta \Delta \Deltasfl q+2k\Gamma2 1 C C C C As0 for any q = 0; 1; : : : and k = 2; 3; : : : : (13) In 1927 P'olya =-=[16]-=- conjectured that Tur'an inequalities hold for the Maclaurin coefficients of F . Precisely, because of (11), P'olya's conjecture reads as (2q + 1)(sb q ) 2 \Gamma (2q \Gamma 1)sb q\Gamma1sb q+1 ? 0; q... |

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Citation Context ... ); (2) where c is real, oes0; m is a nonnegative integer, x k ? 0, and P 1=x k ! 1. If ' is of type I in the Laguerre-P'olya class, we shall write ' 2 L \Gamma P(I). We have adopted the notations of =-=[4, 14, 18]-=- where one may consult for the properties of functions in the Laguerre-Polya class. Generally, L \Gamma P consists of entire functions which are uniform limits on the compact sets of the complex plane... |