In this paper we investigate some properties of generalized Fibonacci and Lucas polynomials. We give some new identities using matrices and Laplace expansion for the generalized Fibonacci and Lucas ...polynomials. Also, we introduce new families of tridiagonal matrices whose successive determinants generate any subsequence of these polynomials.
We investigate hyperderivatives of Drinfeld modular forms and determine formulas for these derivatives in terms of Goss polynomials for the kernel of the Carlitz exponential. As a consequence we ...prove thatv-adic modular forms in the sense of Serre, as defined by Goss and Vincent, are preserved under hyperdifferentiation. Moreover, upon multiplication by a Carlitz factorial, hyperdifferentiation preservesv-integrality.
We consider the sum-of-squares hierarchy of approximations for the problem of minimizing a polynomial
f
over the boolean hypercube
B
n
=
{
0
,
1
}
n
. This hierarchy provides for each integer
r
∈
N
a ...lower bound
f
(
r
)
on the minimum
f
min
of
f
, given by the largest scalar
λ
for which the polynomial
f
-
λ
is a sum-of-squares on
B
n
with degree at most 2
r
. We analyze the quality of these bounds by estimating the worst-case error
f
min
-
f
(
r
)
in terms of the least roots of the Krawtchouk polynomials. As a consequence, for fixed
t
∈
0
,
1
/
2
, we can show that this worst-case error in the regime
r
≈
t
·
n
is of the order
1
/
2
-
t
(
1
-
t
)
as
n
tends to
∞
. Our proof combines classical Fourier analysis on
B
n
with the polynomial kernel technique and existing results on the extremal roots of Krawtchouk polynomials. This link to roots of orthogonal polynomials relies on a connection between the hierarchy of lower bounds
f
(
r
)
and another hierarchy of upper bounds
f
(
r
)
, for which we are also able to establish the same error analysis. Our analysis extends to the minimization of a polynomial over the
q
-ary cube
(
Z
/
q
Z
)
n
. Furthermore, our results apply to the setting of matrix-valued polynomials.
The aim of this paper is to obtain some interesting infinite series representations for the Apostol-type parametrically generalized polynomials with the aid of the Laplace transform and generating ...functions. In particular, by using the method of generating functions, we derive not only recurrence relations, but also several other formulas, identities, and relations as well as combinatorial sums for these parametrically generalized numbers and polynomials and for other known special numbers and polynomials. These identities, relations and combinatorial sums are related to the two-parameter types of the Apostol–Bernoulli polynomials of higher order, the two-parameter types of Apostol–Euler polynomials of higher order, the two-parameter types of Apostol–Genocchi polynomials of higher order, the Apostol–Bernoulli polynomials of higher order, the Apostol–Euler polynomials of higher order, the Apostol–Genocchi polynomials of higher order, the cosine- and sine-Bernoulli polynomials, the cosine- and sine-Euler polynomials, the
λ
-array-type polynomials, the
λ
-Stirling numbers, the polynomials
C
n
x
,
y
, and the polynomials
S
n
x
,
y
. Finally, we present several new recurrence relations for these special polynomials and numbers.
The first aim of this paper is to construct new generating functions for the generalized
λ
-Stirling type numbers of the second kind, generalized array type polynomials and generalized Eulerian type ...polynomials and numbers. We derive various functional equations and differential equations using these generating functions. The second aim is to provide a novel approach to derive identities including multiplication formulas and recurrence relations for these numbers and polynomials using these functional equations and differential equations. Furthermore, we derive some new identities for the generalized
λ
-Stirling type numbers of the second kind, the generalized array type polynomials and the generalized Eulerian type polynomials. We also give many applications related to the class of these polynomials and numbers.
MSC:
12D10, 11B68, 11S40, 11S80, 26C05, 26C10, 30B40, 30C15.
We consider recurrence relations for the polynomials orthonormal with respect to the Sobolev-type inner product and generated by classical orthogonal polynomials, namely: Jacobi polynomials, Legendre ...polynomials, Chebyshev polynomials of the first and the second kind, Gegenbauer (ultraspherical) polynomials, Hermite polynomials.
In 1927, Pólya proved that the Riemann hypothesis is equivalent to the hyperbolicity of Jensen polynomials for the Riemann zeta function ζ(s) at its point of symmetry. This hyperbolicity has been ...proved for degrees d ≤ 3. We obtain an asymptotic formula for the central derivatives ζ
(2n)(1/2) that is accurate to all orders, which allows us to prove the hyperbolicity of all but finitely many of the Jensen polynomials of each degree. Moreover, we establish hyperbolicity for all d ≤ 8. These results follow from a general theorem which models such polynomials by Hermite polynomials. In the case of the Riemann zeta function, this proves the Gaussian unitary ensemble random matrix model prediction in derivative aspect. The general theorem also allows us to prove a conjecture of Chen, Jia, and Wang on the partition function.
Korobov type polynomials are introduced and extensively investigated many mathematicians (\cite{8,9,10,11,12,13,14}). In \ this work, we define unified Apostol Korobov type polynomials and give some ...recurrences relations for these polynomials. Further, we consider the $q$-poly Korobov polynomials and the $q$-poly-Korobov type Changhee polynomials. We give some explicit relations and identities above mentioned functions. KCI Citation Count: 0
Sampling orthogonal polynomial bases via Monte Carlo is of interest for uncertainty quantification of models with random inputs, using Polynomial Chaos (PC) expansions. It is known that bounding a ...probabilistic parameter, referred to as coherence, yields a bound on the number of samples necessary to identify coefficients in a sparse PC expansion via solution to an ℓ1-minimization problem. Utilizing results for orthogonal polynomials, we bound the coherence parameter for polynomials of Hermite and Legendre type under their respective natural sampling distribution. In both polynomial bases we identify an importance sampling distribution which yields a bound with weaker dependence on the order of the approximation. For more general orthonormal bases, we propose the coherence-optimal sampling: a Markov Chain Monte Carlo sampling, which directly uses the basis functions under consideration to achieve a statistical optimality among all sampling schemes with identical support. We demonstrate these different sampling strategies numerically in both high-order and high-dimensional, manufactured PC expansions. In addition, the quality of each sampling method is compared in the identification of solutions to two differential equations, one with a high-dimensional random input and the other with a high-order PC expansion. In both cases, the coherence-optimal sampling scheme leads to similar or considerably improved accuracy.
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