Real Negative Zeros Research Articles

On entire functions from the Laguerre-Polya I class with non-monotonic second quotients of Taylor coefficients

For an entire function $f(z) = \sum_{k=0}^\infty a_k z^k, a_k>0,$ we define its second quotients of Taylor coefficients as $q_k (f):= \frac{a_{k-1}^2}{a_{k-2}a_k}, k \geq 2.$ In the present paper, we study entire functions of order zerowith non-monotonic second quotients of Taylor coefficients. We consider those entire functions for which the even-indexed quotients are all equal and the odd-indexed ones are all equal:$q_{2k} = a>1$ and $q_{2k+1} = b>1$ for all $k \in \mathbb{N}.$We obtain necessary and sufficient conditions under which such functions belong to the Laguerre-P\'olya I class or, in our case, have only real negative zeros. In addition, we illustrate their relation to the partial theta function.

On a partial theta function and its spectrum

The bivariate series defines a partial theta function. For fixed q (∣q∣ < 1), θ(q, ·) is an entire function. For q ∈ (–1, 0) the function θ(q, ·) has infinitely many negative and infinitely many positive real zeros. There exists a sequence of values of q tending to –1+ such that has a double real zero (the rest of its real zeros being simple). For k odd (respectively, k even) has a local minimum (respectively, maximum) at , and is the rightmost of the real negative zeros of (respectively, for k sufficiently large is the second from the left of the real negative zeros of ). For k sufficiently large one has . One has and .

On Power Series having Sections with Multiply Positive Coefficients

Pólya's theorem of 1913 says that if all sections of power series have real negative zeros only, then the series converges in the whole complex plane and its sum satisfies a certain growth condition. Here we show that the assertion of Pólya's theorem remains valid for a much larger class of formal power series and, moreover, a better growth estimate holds.

Totally positive sequences andR-matrix quadratic algebras

Using the theory of Schur functions, we prove that the coefficients of the Hilbert series of a quadraticR-matrix algebra (for a Heckian R-matrix) form a totally positive sequence. The well-known description of the generation series for totally positive sequences implies that the Hilbert series of the quadratic R-matrix algebra is rational with real positive poles and real negative zeros. Using this characterization, we obtain the examples of Koszul and non-R-matrix algebras.

On Power Series Having Sections with Multiply Positive Coefficients and a Theorem of Pólya

Let f ( z ) = ∑ k = 0 ∞ a k z k a 0 > 0 (0.1) be a formal power series. In 1913, G. Pólya [7] proved that if, for all sufficiently large n, the sections f n ( z ) = ∑ k = 0 n a k z k (0.2) have real negative zeros only, then the series (0.1) converges in the whole complex plane C, and its sum f(z) is an entire function of order 0. Since then, formal power series with restrictions on zeros of their sections have been deeply investigated by several mathematicians. We cannot present an exhaustive bibliography here, and restrict ourselves to the references [1, 2, 3], where the reader can find detailed information. In this paper, we propose a different kind of generalisation of Pólya's theorem. It is based on the concept of multiple positivity introduced by M. Fekete in 1912, and it has been treated in detail by S. Karlin [4].

A spread relation for entire functions with negative zeros

Let g g be a canonical product having only real negative zeros and nonintegral order λ \lambda , and let ϕ \phi be the set function defined by 2 π ϕ ( E ) = ∫ E π λ csc ⁡ π λ cos ⁡ λ θ d θ 2\pi \phi (E) = {\smallint _E}\pi \lambda \csc \pi \lambda \cos \lambda \theta d\theta . It is shown that if E ( r ) E(r) is the set of values of θ ∈ ( − π , π ] \theta \in ( - \pi ,\pi ] where | g ( r e i θ ) | ≥ 1 , r n |g(r{e^{i\theta }})| \geq 1,{r_n} is a sequence of Polya peaks of g g and δ \delta is the deficiency of the value zero of g g then ϕ ( E ( r n ) ) ≥ 2 ( 1 − δ ) − 1 \phi (E({r_n})) \geq 2{(1 - \delta )^{ - 1}} . This inequality leads to a sharp spread relation for g g .

On the Zeros of a Class of Bessel Functions Whose Argument and Order are Functions of a Complex Variable

The zeros of a class of modified Bessel functions I (u) and K (u) whose order and argument are analytic functions of the complex variables s are investigated. This is a particular case of more general problems arising when the inverse Laplace transforms of functions describing physical processes are calculated. It is found that the function I (u) has zeros only on the real negative axis in the s-plane, while K (u) can have an infinite set of complex conjugate pairs of zeros in the half-plane Re s<0 together with a finite number of real negative zeros

Rational approximation with real, negative zeros and poles

This note is concerned with the approximation of cosh √ x on [0, 1] by polynomials having only real negative zeros and by rational functions having only real negative zeros and poles. We establish here that cosh √ x can be approximated on [0, 1] by polynomials of degree n having only real negative zeros with an error ⩽ 4 n −1 but not better than c 1 n −1 ( c 1 some positive constant). Further, we show that cosh √ x cannot be approximated on [0, 1 ] by rational functions of total degree n having only real negative zeros and poles with an error better than c 2 n −4.5.

The order of entire functions with radially distributed zeros

It is shown that an entire function with radially distributed zeros has finite order λ \lambda if it has finite lower order μ \mu . It is then shown that functions with real negative zeros only are extremal for the problem of maximizing the Nevanlinna characteristic in the class of entire functions satisfying λ − μ &gt; 1 \lambda - \mu &gt; 1 .

Systematic Classification of Phase Contours and Zeros

A systematic classification of phases contours and zeros is introduced and their effect on asymptotic behavior investigated. A constraint is found on the spacing of closed contours if they are not to contribute to asymptotic behavior. Use is made of a theorem due to P\'olya and Szeg\"o about entire functions with real negative zeros. The concept of topological equivalence of phase contour maps is discussed. The theorem due to P\'olya and Szeg\"o is applied to the fixed-angle scattering amplitude with the assumption that certain kinds of contours or zeros dominate this leads to a constraint on the order of the entire function by which the amplitude may be approximated.

Integral Functions With Negative Zeros

If f(z) is an integral function of non-integral order with only real negative zeros, there is a close connection between the rates of growth of the function and of n (r), the number of zeros of absolute value not exceeding r. The best known theorem is that of Valiron [12], which may be stated as follows.

On Integral Functions with Real Negative Zeros

On Integral Functions with Real Negative Zeros

On Integral Functions with Real Negative Zeros

Proceedings of the London Mathematical SocietyVolume s2-26, Issue 1 p. 185-200 Papers On Integral Functions with Real Negative Zeros E. C. Titchmarsh, E. C. TitchmarshSearch for more papers by this author E. C. Titchmarsh, E. C. TitchmarshSearch for more papers by this author First published: 1927 https://doi.org/10.1112/plms/s2-26.1.185Citations: 22AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volumes2-26, Issue11927Pages 185-200 RelatedInformation