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1.
Infinite Families of Exact Sums of Squares Formulas,Jacobi Elliptic Functions,Continued Fractions,and Schur Functions 总被引:2,自引:0,他引:2
In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi's 4 and 8 squares identities to 4n
2 or 4n(n + 1) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi's explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions. In addition, we extend Jacobi's special analysis of 2 squares, 2 triangles, 6 squares, 6 triangles to 12 squares, 12 triangles, 20 squares, 20 triangles, respectively. Our 24 squares identity leads to a different formula for Ramanujan's tau function (n), when n is odd. These results, depending on new expansions for powers of various products of classical theta functions, arise in the setting of Jacobi elliptic functions, associated continued fractions, regular C-fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. The Schur function form of these infinite families of identities are analogous to the -function identities of Macdonald. Moreover, the powers 4n(n + 1), 2n
2 + n, 2n
2 – n that appear in Macdonald's work also arise at appropriate places in our analysis. A special case of our general methods yields a proof of the two Kac–Wakimoto conjectured identities involving representing a positive integer by sums of 4n
2 or 4n(n + 1) triangular numbers, respectively. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson's C
nonterminating 65 summation theorem, and Andrews' basic hypergeometric series proof of Jacobi's 2, 4, 6, and 8 squares identities. We have (elsewhere) applied symmetry and Schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n
2 and n(n + 1) squares. Our sums of more than 8 squares identities are not the same as the formulas of Mathews (1895), Glaisher (1907), Sierpinski (1907), Uspensky (1913, 1925, 1928), Bulygin (1914, 1915), Ramanujan (1916), Mordell (1917, 1919), Hardy (1918, 1920), Bell (1919), Estermann (1936), Rankin (1945, 1962), Lomadze (1948), Walton (1949), Walfisz (1952), Ananda-Rau (1954), van der Pol (1954), Krätzel (1961, 1962), Bhaskaran (1969), Gundlach (1978), Kac and Wakimoto (1994), and, Liu (2001). We list these authors by the years their work appeared. 相似文献
2.
We use the theory of quadratic base change to derive some new identities involving the Ramanujan -function, and show how the Ramanujan congruence (n) 11(n) (mod 691) follows. 相似文献
3.
Let be the uniform triangulation generated by the usual three-directional mesh of the plane and let 1 be the unit square consisting of two triangles of . We study the space of piecewise polynomial functions in C
k
(R
2) with support 1 having a sufficiently high degree n, which are symmetrical with respect to the first diagonal of 1. Such splines are called 1-splines. We first compute the dimension of this space in function of n and k. Then, for any fixed k0, we prove the existence of 1-splines of class C
k
and minimal degree. These splines are not unique. Finally, we describe an algorithm computing the Bernstein–Bézier coefficients of these splines, and we give an example. 相似文献
4.
Moshe Shaked J. George Shanthikumar 《Annals of the Institute of Statistical Mathematics》1990,42(3):509-531
A collection of random variables {X(), } is said to be parametrically stochastically increasing and convex (concave) in if X() is stochastically increasing in , and if for any increasing convex (concave) function , E(X()) is increasing and convex (concave) in whenever these expectations exist. In this paper a notion of directional convexity (concavity) is introduced and its stochastic analog is studied. Using the notion of stochastic directional convexity (concavity), a sufficient condition, on the transition matrix of a discrete time Markov process {X
n(), n=0,1,2,...}, which implies the stochastic monotonicity and convexity of {X
n(), }, for any n, is found. Through uniformization these kinds of results extend to the continuous time case. Some illustrative applications in queueing theory, reliability theory and branching processes are given.Supported by the Air Force Office of Scientific Research, U.S.A.F., under Grant AFOSR-84-0205. Reproduction in whole or in part is permitted for any purpose by the United States Government. 相似文献
5.
Nikolaos S. Papageorgiou 《Czechoslovak Mathematical Journal》1997,47(3):409-424
In the paper we study the topological structure of the solution set of a class of nonlinear evolution inclusions. First we show that it is nonempty and compact in certain function spaces and that it depends in an upper semicontinuous way on the initial condition. Then by strengthening the hypothesis on the orientor field F(t, x), we are able to show that the solution set is in fact an R
-set. Finally some applications to infinite dimensional control systems are also presented. 相似文献
6.
7.
Let r
k
(n) denote the number of ways n can be expressed as a sum of k squares. Recently, S. Cooper (Ramanujan J. 6:469–490, [2002]), conjectured a formula for r
9(t), t≡5 (mod 8), r
11(t), t≡7 (mod 8), where t is a square-free positive integer. In this note we observe that these conjectures follow from the works of Lomadze (Akad.
Nauk Gruz. Tr. Tbil. Mat. Inst. Razmadze 17:281–314, [1949]; Acta Arith. 68(3):245–253, [1994]). Further we express r
9(t), r
11(t) in terms of certain special values of Dirichlet L-functions. Combining these two results we get expressions for these special values of Dirichlet L-functions involving Jacobi symbols.
相似文献
8.
R. G. Stoneham 《BIT Numerical Mathematics》1970,10(4):481-500
In the computing literature, there are few detailed analytical studies of the global statistical characteristics of a class of multiplicative pseudo-random number generators.We comment briefly on normal numbers and study analytically the approximately uniform discrete distribution or (j,)-normality in the sense of Besicovitch for complete periods of fractional parts {x
0
1
i
/p} on [0, 1] fori=0, 1,..., (p–1)p–1–1, i.e. in current terminology, generators given byx
n+1
1
x
n mod p wheren=0, 1,..., (p–1)p
–1–1,p is any odd prime, (x
0,p)=1,
1 is a primitive root modp
2, and 1 is any positive integer.We derive the expectationsE(X, ),E(X
2, ),E(X
nXn+k); the varianceV(X, ), and the serial correlation coefficient k. By means of Dedekind sums and some results of H. Rademacher, we investigate the asymptotic properties of k for various lagsk and integers 1 and give numerical illustrations. For the frequently used case =1, we find comparable results to estimates of Coveyou and Jansson as well as a mathematical demonstration of a so-called rule of thumb related to the choice of
1 for small k.Due to the number of parameters in this class of generators, it may be possible to obtain increased control over the statistical behavior of these pseudo-random sequences both analytically as well as computationally. 相似文献
9.
We investigate the asymptotic behaviour of the summatory functions of z(n, ), k(n, ) z
(n) and k(n, ) z
(n). 相似文献
10.
Let {p
m
(w)} be the sequence of Jacobi polynomials corresponding to the weightw(x)=(1–x)(1+x), 0, <1. denote=">1.>x
k
(w)=cos
m,k
(w),k=1,...,m, the zeros ofp
m
(w). If +=0, then the estimates
相似文献
11.
A (u1,
u2, . . . )-parking function of length
n is a sequence (x1,
x2, . . . ,
xn)
whose order
statistics (the sequence (x(1),
x(2), . . . ,
x(n)) obtained by rearranging the original sequence in
non-decreasing order) satisfy
x(i)
u(i).
The Gonarov polynomials g
n
(x; a0, a
1, . . . , a
n-1) are
polynomials biorthogonal to the linear functionals (a
i)
Di,
where (a) is evaluation at
a and D
is differentiation. In this paper, we give explicit formulas for the first and second moments of
sums of u-parking functions using Gonarov polynomials by
solving a linear recursion based on a decomposition of the set of sequences of positive integers.
We also give a combinatorial proof of one of the formulas for the expected sum. We specialize
these formulas to the classical case when u
i=a+
(i-1)b and obtain, by
transformations with Abel identities, different but equivalent
formulas for expected sums. These formulas are used to verify the classical case of the
conjecture that the expected sums are increasing functions of the gaps
ui+1
- ui.
Finally, we give analogues of our results for real-valued parking functions.AMS Subject Classification: 05A15, 05A19, 05A20, 05E35. 相似文献
12.
In this paper, we have proven that for the Jordan blockS() withS() (SI),
i=1
n
S() =S()
(n)
(n 1) has unique finite (SI) decomposition up to a similarity. As result, we obtain that ifV is a Volterra operator onH=L
2([0, 1]), thenV
(n)
has unique finite (SI) decomposition.This project was supported by National Natural Science Foundation of China. 相似文献
13.
In this work the Dirichlet series
associated with real strongly q-multiplicative functions f(n) are studied. We will confine ourselves to the case
i=0
q–1
f(i) = 0. It is known that in this case the function
f
(s) has an analytic continuation to the whole complex plane as an entire function with trivial zeros on the negative real line. The real function
f
(t) satisfying the integral equation with delayed argument
for some nonzero real
f
naturally appears in the representation of the function
f
(s). In this article we find some asymptotic properties of the function
f
(s), prove that
f
(s) is an entire function of order 2, and also prove that in the region
the function
f
(s) has only trivial zeros which are simple. 相似文献
14.
By means of Jacobi?s triple product identity and the t -coefficient method, we establish a general series expansion formula with five free parameters for the product of arbitrary two Jacobi theta functions. It embodies the triple, quintuple, sextuple and septuple theta function product identities and the generalized Schröter formula. As further applications, we also set up a series expansion formula for the product of three theta functions. It not only generalizes Ewell?s and Chen–Chen–Huang?s octuple product identities, but also contains three cubic theta function identities due to Farkas–Kra and Ramanujan respectively and the Macdonald identity for the root system A2 as special cases. In the meantime, many other new identities including a new short expression of the triple theta series of Andrews are also presented. 相似文献
15.
R. Nair 《Monatshefte für Mathematik》1998,125(3):241-253
Supposek
n denotes either (n) or (p
n) (n=1,2,...) where the polynomial maps the natural numbers to themselves andp
k denotes thek
th rationals prime. Also let
denote the sequence of convergents to a real numberx and letc
n(x))
n=1
be the corresponding sequence of partial quotients for the nearest integer continued fraction expansion. Define the sequence of approximation constants
n(x))
n=1
by
16.
Let f(a, b) denote Ramanujan's symmetric theta function. In his Lost Notebook, Ramanujan claimed that the circular summation of n-th powers of f satisfies a factorization of the form f(a, b)F(ab). He listed elegant identities for n = 2, 3, 4, 5 and 7. We present alternative proofs of his claims. 相似文献
17.
Let w() be a positive weight function on the unit circle of the complex plane. For a sequence of points {
k
}
k = 1
included in a compact subset of the unit disk, we consider the orthogonal rational functions
n
that are obtained by orthogonalization of the sequence { 1, z / 1, z
2 / 2, ... } where
, with respect to the inner product
In this paper we discuss the behaviour of
n
(t) for t = 1 and n under certain conditions. The main condition on the weight is that it satisfies a Lipschitz–Dini condition and that it is bounded away from zero. This generalizes a theorem given by Szeg in the polynomial case, that is when all
k
= 0. 相似文献
18.
In the present paper we consider a q-analog of t–(v,k,)-designs. It is canonic since it arises by replacing sets by vector spaces over GF(q), and their orders by dimensions. These generalizations were introduced by Thomas [Geom.Dedicata vol. 63, pp. 247–253 (1996)] they are called t –(v,k,;q)- designs. A few of such q-analogs are known today, they were constructed using sophisticated geometric arguments and case-by-case methods. It is our aim now to present a general method that allows systematically to construct such designs, and to give complete catalogs (for small parameters, of course) using an implemented software package. In order to attack the (highly complex) construction, we prepare them for an enormous data reduction by embedding their definition into the theory of group actions on posets, so that we can derive and use a generalization of the Kramer-Mesner matrix for their definition, together with an improved version of the LLL-algorithm. By doing so we generalize the methods developed in a research project on t –(v,k,)-designs on sets, obtaining this way new results on the existence of t–(v,k,;q)-designs on spaces for further quintuples (t,v,k,;q) of parameters. We present several 2–(6,3,;2)-designs, 2–(7,3,;2)-designs and, as far as we know, the very first 3-designs over GF(q).classification 05B05 相似文献
19.
We derive a new formula for the supersymmetric Schur polynomial s
(x/y). The origin of this formula goes back to representation theory of the Lie superalgebra gl(m/n). In particular, we show how a character formula due to Kac and Wakimoto can be applied to covariant representations, leading to a new expression for s
(x/y). This new expression gives rise to a determinantal formula for s
(x/y). In particular, the denominator identity for gl(m/n) corresponds to a determinantal identity combining Cauchy's double alternant with Vandermonde's determinant. We provide a second and independent proof of the new determinantal formula by showing that it satisfies the four characteristic properties of supersymmetric Schur polynomials. A third and more direct proof ties up our formula with that of Sergeev-Pragacz. 相似文献
20.
In this paper we present a study of spaces of splines in C
k
(R
2) with supports the square 1 and the lozenge 1 formed respectively by four and eight triangles of the uniform four directional mesh of the plane. Such splines are called 1 and 1-splines. We first compute the dimension of the space of 1-splines. Then we prove the existence of a unique 1-spline of minimal degree for any fixed k0. By using this last result, we also prove the existence of a unique 1-spline of minimal degree. Finally, we describe algorithms allowing to compute the Bernstein–Bézier coefficients of 1-spline and 1-spline of minimal degree. 相似文献
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