scholarly journals Note on a special determinant

1937 ◽  
Vol 30 ◽  
pp. xxvii-xviii
Author(s):  
A. C. Aitken
Keyword(s):  
Power Series ◽  
Convergent Power Series ◽  
Image Position

Suppose a polynomial or convergent power seriesis raised to powers j = 0, 1, 2, 3, … The coefficients of xk in [f(x)]j, k = 0, 1, 2, …, may be entered as elements in positions (j, k) in an array or matrix F, thus:By construction all elements in column (k) have weight (sum of suffixes) equal to k.

On the Taylor coefficients of rational functions

1956 ◽  
Vol 52 (1) ◽  
pp. 39-48 ◽  
Author(s):  
K. Mahler ◽  
J. W. S. Cassels
Keyword(s):  
Power Series ◽  
Rational Function ◽  
Rational Functions ◽  
Partial Solution ◽  
Convergent Power Series ◽  
Taylor Coefficients ◽  
Image Position

Let F(z) be a rational function of z which is regular at z = 0 and so possesses a convergent power seriesThe problem arises of characterizing those rational functions F(z) that have infinitely many vanishing Taylor coefficientsfh. After earlier and more special results by Siegel(2) and Ward(4) I applied in 1934(1) a p-adic method due to Skolem(3) to the problem and obtained the following partial solution.

Analogue of a theorem of Khintchine in fields of formal power series

1966 ◽  
Vol 62 (4) ◽  
pp. 637-642 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Power Series ◽  
Positive Constant ◽  
Classical Result ◽  
Real Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
The Difference ◽  
Image Position ◽  
Formal Power ◽  
Positive Integers

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.

scholarly journals An algebraically closed field

1968 ◽  
Vol 9 (2) ◽  
pp. 146-151 ◽  
Author(s):  
F. J. Rayner
Keyword(s):  
Positive Integer ◽  
Power Series ◽  
Formal Power Series ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Zero Characteristic ◽  
Image Position ◽  
Formal Power ◽  
Puiseux Expansions

Letkbe any algebraically closed field, and denote byk((t)) the field of formal power series in one indeterminatetoverk. Letso thatKis the field of Puiseux expansions with coefficients ink(each element ofKis a formal power series intl/rfor some positive integerr). It is well-known thatKis algebraically closed if and only ifkis of characteristic zero [1, p. 61]. For examples relating to ramified extensions of fields with valuation [9, §6] it is useful to have a field analogous toKwhich is algebraically closed whenkhas non-zero characteristicp. In this paper, I prove that the setLof all formal power series of the form Σaitei(where (ei) is well-ordered,ei=mi|nprt,n∈ Ζ,mi∈ Ζ,ai∈k,ri∈ Ν) forms an algebraically closed field.

Analytic functions of finite dimensional linear transformations

1959 ◽  
Vol 55 (1) ◽  
pp. 51-61 ◽  
Author(s):  
S. N. Afriat
Keyword(s):  
Power Series ◽  
Integral Formula ◽  
Characteristic Root ◽  
Interpolation Formula ◽  
General Function ◽  
Linear Transformations ◽  
Multiple Characteristic ◽  
Characteristic Roots ◽  
Image Position ◽  
Necessary And Sufficient

Since the first introduction of the concept of a matrix, questions about functions of matrices have had the attention of many writers, starting with Cayley(i) in 1858, and Laguerre(2) in 1867. In 1883, Sylvester(3) defined a general function φ(a) of a matrix a with simple characteristic roots, by use of Lagrange's interpolation formula, and Buchheim (4), in 1886, extended his definition to the case of multiple characteristic roots. Then Weyr(5) showed in 1887 that, for a matrix a with characteristic roots lying inside the circle of convergence of a power series φ(ζ), the power series φ(a) is convergent; and in 1900 Poincaré (6) obtained the formulaefor the sum, where C is a circle lying in and concentric with the circle of convergence, and containing all the characteristic roots in its ulterior, such a formula having effectively been suggested by Frobenius(7) in 1896 for defining a general function of a matrix. Phillips (8), in 1919, discovered the analogue, for power series in matrices, of Taylor's theorem. In 1926 Hensel(9) completed the result of Weyr by showing that a necessary and sufficient condition for the convergence of φ(a) is the convergence of the derived series φ(r)(α) (0 ≼ r < mα; α) at each characteristic root α of a, of order r at most the multiplicity mα of α. In 1928 Giorgi(10) gave a definition, depending on the classical canonical decomposition of a matrix, which is equivalent to the contour integral formula, and Fantappie (11) developed the theory of this formula, and obtained the expressionfor the characteristic projectors.

Chromatic Solutions, II

1982 ◽  
Vol 34 (4) ◽  
pp. 952-960 ◽  
Author(s):  
W. T. Tutte
Keyword(s):  
Power Series ◽  
Research Report ◽  
Chromatic Polynomials ◽  
Image Position

This paper is a continuation of the Waterloo Research Report CORR 81-12, (see [1]) referred to in what follows as I. That Report is entitled “Chromatic Solutions”. It is largely concerned with a power series h in a variable z2, in which the coefficients are polynomials in a “colour number” λ. By definition the coefficient of z2r, where r > 0, is the sum of the chromatic polynomials of the rooted planar triangulations of 2r faces. (Multiple joins are allowed in these triangulations.) Thus for a positive integral λ the coefficient is the number of λ-coloured rooted planar triangulations of 2r faces. The use of the symbol z2 instead of a simple letter t is for the sake of continuity with earlier papers.In I we consider the case(1)where n is an integer exceeding 4.

On the Distribution of Sum of Independent Positive Binomial Variables

1970 ◽  
Vol 13 (1) ◽  
pp. 151-152 ◽  
Author(s):  
J. C. Ahuja
Keyword(s):  
Power Series ◽  
Inversion Formula ◽  
Probability Function ◽  
Random Variables ◽  
Random Variable ◽  
Characteristic Functions ◽  
Binomial Probability ◽  
Generalized Power Series ◽  
Alternative Derivation ◽  
Image Position

Let X1, X2, …, Xn be n independent and identically distributed random variables having the positive binomial probability function1where 0 < p < 1, and T = {1, 2, …, N}. Define their sum as Y=X1 + X2 + … +Xn. The distribution of the random variable Y has been obtained by Malik [2] using the inversion formula for characteristic functions. It appears that his result needs some correction. The purpose of this note is to give an alternative derivation of the distribution of Y by applying one of the results, established by Patil [3], for the generalized power series distribution.

scholarly journals The exponent of certain finite p-groups

1990 ◽  
Vol 33 (3) ◽  
pp. 483-490 ◽  
Author(s):  
I. O. York
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Formal Power Series ◽  
Group Operation ◽  
Image Position ◽  
Formal Power ◽  
Quotient Groups

In this paper, for R a commutative ring, with identity, of characteristic p, we look at the group G(R) of formal power series with coefficients in R, of the formand the group operation being substitution. The results obtained give the exponent of the quotient groups Gn(R) of this group, n∈ℕ.

Chromatic Solutions

1982 ◽  
Vol 34 (3) ◽  
pp. 741-758 ◽  
Author(s):  
W. T. Tutte
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Chromatic Polynomial ◽  
Basic Theory ◽  
Root Vertex ◽  
Independent Variables ◽  
Image Position ◽  
Formal Power ◽  
Two Independent Variables

Early in the Seventies I sought the number of rooted λ-coloured triangulations of the sphere with 2p faces. In these triangulations double joins, but not loops, were permitted. The investigation soon took the form of a discussion of a certain formal power series l(y, z, λ) in two independent variables y and z.The basic theory of l is set out in [1]. There l is defined as the coefficient of x2 in a more complicated power series g(x, y, z, λ). But the definition is equivalent to the following formula.1Here T denotes a general rooted triangulation. n(T) is the valency of its root-vertex, and 2p(T) is the number of its faces. P(T, λ) is the chromatic polynomial of the graph of T.

scholarly journals Polynomially bounded multisequences and analytic continuation

1982 ◽  
Vol 23 (1) ◽  
pp. 41-52
Author(s):  
Daniel J. Troy
Keyword(s):  
Power Series ◽  
Analytic Continuation ◽  
Linear Functional ◽  
Necessary And Sufficient Condition ◽  
Convergence Criteria ◽  
Dimensional Torus ◽  
Unit Factor ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Class Of Functions

Given a polynomially bounded multisequence {fm}, where m = (m1, …, mk) ∈ ℤk, we will consider 2k power series in exp(iz1), …, exp(izk), each representing a holomorphic function within its domain of convergence. We will consider this same multisequence as a linear functional on a class of functions defined on the k-dimensional torus by a Fourier series, , with the proper convergence criteria. We shall discuss the relationships that exist between the linear functional properties of the multisequence and the analytic continuation of the holomorphic functions. With this approach we show that a necessary and sufficient condition that the multisequence be given by a polynomial is that each of the power series represents, up to a unit factor, the same function that is entire in the variables

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