This paper deals with Deligne's conjecture on the critical values of L-functions. Let Z G⊗h (s) denote the tensor product L-function attached to a Siegel modular form G of weight k and an elliptic cusp form h of weight l. We assume that the first Fourier-Jacobi coefficient of G is not identically zero. Then Deligne's conjecture is fully proven for Z G⊗h (s), when l ≤ 2k − 2 and partly for the remaining case.
Sym 2 M k+2ν is given. This leads to a basic characterization of the Spezialschar property. The results of this paper are directly related to the non-vanishing of certain special values of L-functions related to the Gross-Prasad conjecture. This is illustrated by a significant example in the paper.
In this paper we show that, in the holomorphic automorphic case, the Borcherds lifts on the orthogonal group O(2, n + 2) are characterized by certain symmetries and describe the inverse of the Borcherds lifting in terms of Fourier-Jacobi expansion. We also give a characterization of the modular polynomials by certain symmetries.
The Nekrasov-Okounkov hook length formula provides a fundamental link between the theory of partitions and the coefficients of powers of the Dedekind eta function. In this paper we examine three conjectures presented by Amdeberhan. The first conjecture is a refined Nekrasov-Okounkov formula involving hooks with trivial legs. We prove the conjecture. The second conjecture is on properties of the roots of the underlying D'Arcais polynomials. We give a counterexample and present a new conjecture. The third conjecture is on the unimodality of the coefficients of the involved polynomials. We confirm the conjecture up to the polynomial degree 1000. 1 24 ∞ m=1 (1 − q m ) (see [On03]).
In this paper, we investigate a generalization of the Bessenrodt–Ono inequality by following Gian–Carlo Rota’s advice in studying problems in combinatorics and number theory in terms of roots of polynomials. We consider the number of k-colored partitions of n as special values of polynomials $$P_n(x)$$
P
n
(
x
)
. We prove for all real numbers $$x >2 $$
x
>
2
and $$a,b \in \mathbb {N}$$
a
,
b
∈
N
with $$a+b >2$$
a
+
b
>
2
the inequality: $$\begin{aligned} P_a(x) \, \cdot \, P_b(x) > P_{a+b}(x). \end{aligned}$$
P
a
(
x
)
·
P
b
(
x
)
>
P
a
+
b
(
x
)
.
We show that $$P_n(x) < P_{n+1}(x)$$
P
n
(
x
)
<
P
n
+
1
(
x
)
for $$x \ge 1$$
x
≥
1
, which generalizes $$p(n) < p(n+1)$$
p
(
n
)
<
p
(
n
+
1
)
, where p(n) denotes the partition function. Finally, we observe for small values, the opposite can be true, since, for example: $$P_2(-3+ \sqrt{10}) = P_{3}(-3 + \sqrt{10})$$
P
2
(
-
3
+
10
)
=
P
3
(
-
3
+
10
)
.
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