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Articles 331 - 351 of 351
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On Certain Arithmetical Functions Due To G. Humbert, M. A. Basoco
On Certain Arithmetical Functions Due To G. Humbert, M. A. Basoco
Department of Mathematics: Faculty Publications
G. Humbert has discussed, in a series of brief notes, a certain class of entire functions with interesting arithmetical properties. These functions are defined, in an essentially unique manner, by certain functional equations. The Fourier series representations of the solutions of these equations are similar in form to those for the elliptic functions snu, cnu, dnu, and so on. They differ from these, however, in that their domain of validity extends throughout the entire complex plane (Z∞ excluded) and moreover, in that their arithmetized forms involve incomplete numerical functions of the divisors of an integer.
In the …
A Theorem On The Unit Groups Of Simple Algebras, Ralph Hull
A Theorem On The Unit Groups Of Simple Algebras, Ralph Hull
Department of Mathematics: Faculty Publications
Let k be an algebraic number field of finite degree m and let A be a normal simple algebra of degree n, order n2, over k. Our object is to prove the following theorem.
THEOREM. If A is an R-algebra, that is, if n>2 or at least one infinite prime place of k is unramified in A when n=2, then any two distinct maximal orders of A have distinct groups of units.
On The Fourier Developments Of A Certain Class Of Theta Quotients, M. A. Basoco
On The Fourier Developments Of A Certain Class Of Theta Quotients, M. A. Basoco
Department of Mathematics: Faculty Publications
In this paper we shall be concerned with the functions φka(z) defined by the relation
(1) φka(z) ≡{d/(dz)logℓα (z,q)}k = {[ℓά(z,q)] ÷ [ℓα(z,q)]k, α + 0, 1, 2, 3,
where ℓα(z, q) is a Jacobi theta function and k is a positive integer. In the first place, we shall derive the Fourier developments which represent these functions in a certain strip of the complex plane; it …
The Basic Analogue Of Kummer’S Theorem, J. A. Daum
The Basic Analogue Of Kummer’S Theorem, J. A. Daum
Department of Mathematics: Faculty Publications
About one hundred years ago, E. E. Kummer proved the formula
(1) 2F1 [1 + a – b a, b; -1] = [Γ(1 + a - b)Γ(1 + a/2) ÷ Γ(1+a)Γ(1+a/2 - b)
which has since been known as Kummer's theorem. This appears to be the simplest relation involving a hypergeometric function with argument ( - 1).
All the relations in the theory of hypergeometric series rF8 which have analogues in the theory of basic serie3 are those in which the argument is ( + 1 ) …
On Certain Basic Series, John Daum
On Certain Basic Series, John Daum
Department of Mathematics: Faculty Publications
The identity
(1) Σn=1∞ (qn)/[(1-qn)2] {(1÷(1-q)) + (1÷(1-q2)) + … + (1 ÷ (1 -qn))} = Σn=1∞ [(n2qn)
was deduced from arithmetical considerations by E. T. Bell. About five years ago, W. N. Bailey proved the relation
(2) Σn=0∞ [(1-q)(1-q2)…(1-qn] ÷ [(1-z)(1-qz)…(1-qnz)] x [(zn + 1) ÷ (1 - qn …
Orthogonal Polynomials With Orthogonal Derivatives, M. S. Webster
Orthogonal Polynomials With Orthogonal Derivatives, M. S. Webster
Department of Mathematics: Faculty Publications
We are concerned with the following assertion:
THEOREM. If {φn(x)} and { φn’(x)} are orthogonal systems of polynomials, then {φn(x)} may be reduced to the classical polynomials of Jacobi, Laguerre, or Hermite by means of a linear transformation on x.
Note On The Greatest Integer Function, M. A. Basoco
Note On The Greatest Integer Function, M. A. Basoco
Department of Mathematics: Faculty Publications
In this note we wish to record certain finite sums involving the greatest integer function E(x), which seem to be of some interest. Hermite* has shown that the generating function for E(x) has the simple form, (1) xb ÷(1 -x)(1 -xa) = Σ(n)E[(n + a - b) ÷ a]xn, where a, b are positive integers. To him is due, likewise, the development (2) ) xb ÷(1 -x)(1 + xa) =Σ(n)E1[( …
On The Summability Of A Certain Class Of Series Of Jacobi Polynomials, A. P. Cowgill
On The Summability Of A Certain Class Of Series Of Jacobi Polynomials, A. P. Cowgill
Department of Mathematics: Faculty Publications
The result obtained in this paper is as follows:
The series Σni[((p + 1)(p +3)…(p +2n -1)) ÷(2nn!) X((p-1)/2)n (x), where Xn(p-1)/2(x) (hereafter indicated simply by Xn) is a symmetric Jacobi polynomial p >-1, and i a positive integer, is summable (C, k),k>i—1/2, for the range -1 <x<1.
On The Summability And Generalized Sum Of A Series Of Legendre Polynomials, W. C. Brenke
On The Summability And Generalized Sum Of A Series Of Legendre Polynomials, W. C. Brenke
Department of Mathematics: Faculty Publications
The results obtained in this paper are as follows. (A) The series of Legendre polynomials ΣnpXn(x), where p is a positive integer, is summable (H, p) for -1< x < 1, and summable (H, p +1) for - 1 ≤ x < 1 .
On The Element Of Decomposition Of A Doubly Periodic Function Of The Second Kind, M. A. Basoco
On The Element Of Decomposition Of A Doubly Periodic Function Of The Second Kind, M. A. Basoco
Department of Mathematics: Faculty Publications
Hermite has shown that a meromorphic function which satisfies periodicity relations of the form (1) F (z + 2ω) = μF (z), F (z + 2ω’) = μ’F (z), where ω’/ω = a+ib, b>0, and μ, μ' are independent of z, maybe expressed in terms of the function v{z + X) (2) G(z) = σ(z + λ) ÷ σ(z) + σ(λ) eρz, and its derivatives, in which λ, ρ are suitably determined constants and σ(u) is the …
On The Trigonometric Developments Of Certain Doubly Periodic Functions Of The Second Kind, M. A. Basoco
On The Trigonometric Developments Of Certain Doubly Periodic Functions Of The Second Kind, M. A. Basoco
Department of Mathematics: Faculty Publications
The class of meromorphic functions which satisfy periodicity relations of the form (1) ƒ(z + 2ωl) = c1f(z), f(z + 2ω2) = c2f{z), where the multipliers c1 and c2 are independent of z, and ωl/ω2 is a complex number with non-vanishing imaginary part, has been named by Hermite doubly periodic of the second kind. It is possible to make the study of these functions depend on others of the same type, but such that one of the multipliers, say cl, is unity. In what follows …
The Practical Evaluation Of Resultants, T. A. Pierce
The Practical Evaluation Of Resultants, T. A. Pierce
Department of Mathematics: Faculty Publications
The purpose of the present note is to give a practical method of evaluating the resultant of two equations. The method is particularly effective when the degree of one of the equations is high while that of the other is low. Use will be made of certain results in the theory of matrices.
On The Trigonometric Expansion Of Elliptic Functions, M. A. Basoco
On The Trigonometric Expansion Of Elliptic Functions, M. A. Basoco
Department of Mathematics: Faculty Publications
The problem of expressing an elliptic function in terms of infinite sums of trigonometric functions has been treated by Hermite, Briot and Bouquet, A. C. Dixon and others. In the present paper we treat the same problem from the point of view of Cauchy's residue theorem in function theory, which is also Briot and Bouquet's starting point, but we differ from these authors in that the integrand we use leads to an expansion for an elliptic function which is valid in an arbitrarily wide, but finite, strip of the complex plane, and which contains certain classical results as special cases. …
Parametric Solutions Of Certain Diophantine Equations, T. A. Pierce
Parametric Solutions Of Certain Diophantine Equations, T. A. Pierce
Department of Mathematics: Faculty Publications
In this note parametric solutions of certain diophantine equations are given. The method of obtaining the solutions is derived from an equation involving the determinants of certain matrices. It will be recognized that the method is a generalization of the method of Euler and Lagrange which depends on forms which repeat under multiplication. The matrices used in this paper must be such that their forms are retained under matric multiplication and addition. When integer values are assigned to the parameters of our solutions we obtain integer solutions of the particular equation under consideration; however not all integer solutions are necessarily …
A Certain Multiple-Parameter Expansion, H. P. Doole
A Certain Multiple-Parameter Expansion, H. P. Doole
Department of Mathematics: Faculty Publications
C. C. Camp has shown the convergence of the expansion of an arbitrary function in terms of the solutions of the systems of equations
X1’ (λa1 - Σi=2nμi)X1 = 0,
X1’ (λai + μi)Xi = 0, (j = 2, 3, …, n),
where the ai’s are functions of x, with the boundary conditions
Xi(-π) = Xi(π), (j = 1, 2, …, n).
In this paper it is intended to use a …
On Polynomial Solutions Of A Class Of Linear Differential Equations Of The Second Order, W. C. Brenke
On Polynomial Solutions Of A Class Of Linear Differential Equations Of The Second Order, W. C. Brenke
Department of Mathematics: Faculty Publications
Certain well known polynomials have a number of common properties. They arise as coefficients of tn in the expansion of a generating function ; they may be obtained by means of orthogonalization of a set of functions xⁿg(x) when the function ρ(x) = g2(x) and the interval are properly chosen ; they may be regarded as polynomials which become orthogonal when multiplied by a proper factor g(x) ; they satisfy a certain type of difference equation ; they satisfy a certain type of differential equation. The results …
Matrices Whose Characteristic Equations Are Cyclic, T. A. Pierce
Matrices Whose Characteristic Equations Are Cyclic, T. A. Pierce
Department of Mathematics: Faculty Publications
One of Sylvester's theorems f on matrices states that if the characteristic equation
(1) | M - λI| = f(λ) = 0
of a square matrix M has the roots λ1, λ2, … , λn, then the characteristic equation
(2) | φM - ρI| = = g(ρ) = 0
of any integral function of M, namely, φM, has the roots ρi = φ (λi), i = 1, 2, … , n. In this note an isomorphism is shown to exist between …
Symmetric Functions Of N-Ic Residues (Mod P), T. A. Pierce
Symmetric Functions Of N-Ic Residues (Mod P), T. A. Pierce
Department of Mathematics: Faculty Publications
If p be an odd prime, q is said to be an n- ic residue of p if the congruence xn = q (mod p) has solutions; otherwise q is an n-ic non-residue of p. A necessary and sufficient condition that q be an n-ic residue of p is that
(1) q (p- 1)/δ ≡ 1, (mod p),
where δ = g.c.d. (p - 1, n). The numberf of n-ic residues of a given prime p is (p-1)/δ.
It is with the symmetric functions of these …
An Approximation To The Least Root Of A Cubic Equation With Application To The Determination Of Units In Pure Cubic Fields, T. A. Pierce
An Approximation To The Least Root Of A Cubic Equation With Application To The Determination Of Units In Pure Cubic Fields, T. A. Pierce
Department of Mathematics: Faculty Publications
Approximation to the Least Root of a Cubic Bernoulli's method of approximating the largest root of an equation
(1) x3 = ax2 + bx + c,
with real coefficients, is to use (1) as a scale of relation for the recursion formula An = aAn-1 + bAn-2 + cAn-3. Successive A’s are calculated starting from any initial values. Then An+i/An for increasing values of n approximates that root of (1) which has the greatest absolute value if that root is real. The method here given for …
A Set Of Completely Independent Postulates For The Linear Order Η*, M. G. Gaba
A Set Of Completely Independent Postulates For The Linear Order Η*, M. G. Gaba
Department of Mathematics: Faculty Publications
Professor E. V. Huntington has published three sets of completely independent postulates for serial order. His set A involves four postulates, which is as high a number of postulates as had been proved completely independent. In the present paper are given seven postulates which form a categorical and completely independent set for the linear order.
Our basis is a class of elements [p] and an undefined dyadic relation (called 'less than') among the elements. If we are given two elements p1p2 and if the relation p1 less than p2 holds, we will symbolize …
A Theorem On Semi-Continuous Functions, Henry Blumberg
A Theorem On Semi-Continuous Functions, Henry Blumberg
Department of Mathematics: Faculty Publications
Recently G. C. Young and A. Denjoy have communicated theorems—those in Denjoy's memoir are of an especially comprehensive character—dealing, in particular, with point sets where the four derivatives of a given continuous function are identical. It is the purpose of this note to treat an analogous problem that arises when "derivative" is replaced by "saltus." However, instead of confining ourselves to "saltus," we prove a more general theorem that applies essentially to all semi-continuous functions. We preface the proof of this theorem with the following