We prove that the order derivative of Bessel function J_ν (z) with any real ν has infinitely many positive real zeros and non-real zeros in the left half-plane. In the case ν≥ 0, it is further shown that the order derivative has no non-real zeros in the right half-plane and all of its zeros in the left half-plane lie in a ν -independent horizontal strip.
This paper investigates the Hausdorff dimension of exceptional sets arising from the growth behavior of non-decreasing partial quotients in continued fractions. Let x∈ (0,1) be an irrational with continued fraction expansion [a_1(x),a_2(x),a_3(x),… ] . For a given function ψ : ℕ→ℝ^+ satisfying ψ (n)→∞ as n→∞ , we consider the sets of points for which a_n+1(x) ≥ a_n(x) holds for all n∈ℕ and log a_n(x) grows asymptotically like ψ (n) in the sense of limsup, liminf, and the ordinary limit. Previous results in this direction were obtained under the assumption that the limit lim _n→∞ψ (n)/log n exists. In this paper, we remove this assumption. For the limsup and liminf growth sets, we prove that the Hausdorff dimension is completely determined by the lower and upper limits of ψ (n)/log n , respectively, and we establish explicit and symmetric dimension formulas. In contrast, for the ordinary limit growth set, we show that the Hausdorff dimension depends on the limiting behavior of ψ (n)/log n , and in general cannot be characterized solely by its lower or upper limits.
In this paper, we obtain several new q-supercongruences from Gasper and Rahman’s quadratic summation, which include a confirmation of a recent conjecture proposed by Guo and Zhao. In particular, we prove the following supercongruence: for any positive integer r and any prime p≡ 1 4 with p ≥ 13 , ∑ _k=0^(3p^r+1)/4(6k+1)( 1/2) _k^3( -1/4) _k( 1/4) _k/k!^3(k+1)!( -1/2) _k≡ 0 p^4r, where the shifted factorial (x)_n is defined by (x)_n=x(x+1)(x+2)⋯ (x+n-1) for n∈ℤ^+ and (x)_0=1 .
Let ϑ _j(τ ) , j=2,3,4 , be the Jacobi theta functions. For a positive integer r, let α _r:=ϑ _2(rτ )^4/ϑ _3(rτ )^4 . An algebraic relation between α _1 and α _r is called a modular equation of degree r. In this paper, we introduce some new ideas for computing modular equations.
Solutions of the Gauss hypergeometric equation, two of whose characteristic exponent differences are half-odd-integers and the third is an arbitrary complex number, are called dihedral Gauss hypergeometric functions. They can be expressed in terms of elementary functions since they belong to Schwarz’s dihedral class. In this article, explicit trigonometric and algebraic identities for dihedral Gauss hypergeometric functions in terms of the Jacobi polynomials with some pre-factors are presented. To this end, four different sets of linearly independent solutions of a specific ordinary differential equation are obtained where the first three sets contain dihedral Gauss hypergeometric functions and the fourth one consists of elementary functions. Dihedral identities are then obtained by expressing the hypergeometric solutions as linear combinations of the elementary ones. Moreover, dihedral identities are extended to a more general class of Gauss hypergeometric functions.
A multiplicative arithmetic function f is called strongly multiplicative if f(p^n)=f(p) for every prime number p and every positive integer n. We say that an (a, b)-Fibonacci sequence is unitary if a + b=1 . A totient f is an enumerative totient if the Dirichlet inverse f^-1 is strongly multiplicative, and for example Euler’s totient is enumerative. We show a one-to-one correspondence between strongly multiplicative functions and prime-indexed families of unitary Fibonacci sequences. These two mathematical objects are equivalent in the sense that each can be reconstructed from the other. Some characterizations, properties, and examples link the three mathematical objects in the paper’s title.
We study antiderivatives of modular integrals for the groups Γ _0^+(p) and their Mellin transforms where p∈{1,2,3} . In this paper, we show that these antiderivatives satisfy certain transformation laws involving logarithmic terms determined by rational period functions. Furthermore, we establish the analytic properties of the associated Mellin transform and derive its functional equation.
In this article, we aim to provide a series of very general identities, which arose from the results obtained by Calkin, Hirschhorn, and Zhang between 1994 and 2008. Our approach differs completely as our method does not use the Omega operator. As one of the main consequences, we also obtain a general identity recovering Hirschhorn’s classic identity and its alternating variant.
Let G=C_5× C_5 , and let S(G) denote the set of integer values of its group determinant. Previous work determines the values in S(G) coprime to 5 and proves that every 5-divisible value is divisible by 5^8 . We prove the converse inclusion 5^8 ℤ⊆ S(G). Consequently, S(G)={m∈ℤ:m≡± 1 or ± 7 25}∪ 5^8ℤ . The proof uses a general shift criterion and three explicit polynomials whose group determinants are 5^8 , 2· 5^8 , and 5^9 .
In this paper, we study a class of partitions called unrestricted singular overpartitions with parameters k and i , which is an overpartition such that no part is divisible by k, and any single occurrence of a part congruent to ± i k may be overlined. The number of all such overpartitions of n is denoted by Cu_k,i(n) . Previous studies have established several congruence properties for Cu_3,1(n) modulo 2 and 4, and Cu_4,1(n) modulo 2, 4, and 8. We extend these results by deriving new congruences for Cu_6,1(n) modulo 36, 48, and 64. In addition, we establish connections between unrestricted singular overpartitions and other partition families, including (k,ℓ ) -regular bipartitions and overpartitions with ℓ -regular nonoverlined parts.
The divisor function d(n), which counts the divisors of the integer n, over arithmetic progressions has been widely studied. We consider the divisor sum related to the Piatetski-Shapiro sequence 𝒩^(c) :=(⌊ n^c⌋ )_n=1^∞ in arithmetic progressions, namely ∑ _[ n ⩽ x, n ∈𝒩^(c); n ≡ a q ] d(n). We obtain an asymptotic formula with a main term and an error term for 1< c < 6/5 and q ≪ x^3/5c-1/2-ε . The range of c is the same as the previous result related to the divisor sum over Piatetski-Shapiro sequences. A key idea is that we have a new bound on exponential sums over arithmetic progressions. As an application, we also extend the admissible range of c for Piatetski-Shapiro primes in arithmetic progressions.
Abstract We give closed formulas for the first few expansion coefficients of the basic modular forms for $${{\,\textrm{GL}\,}}(r, \mathbb {F}_{q}[T])$$ GL ( r , F q [ T ] ) . Here the rank $$r$$ r is larger or equal to $$3$$ 3 , and the forms in question include the coefficient forms $$g_{1}, \dots , g_{r}$$ g 1 , ⋯ , g r and the Eisenstein series $$E_{q^{i}-1}$$ E q i - 1 ( $$i \in \mathbb {N}$$ i ∈ N ).
Let α be an irrational number, β and c be real numbers. The Piatetski–Shapiro sequence and the corresponding non-homogeneous Beatty sequence are defined as 𝒩^(c)=(⌊ n^c⌋ )_n=1^∞ (c>1, c ∉ℕ), and ℬ_α , β=(⌊α n+β⌋ )_n=1^∞ respectively. For every R > 1 , we say that a natural number is an R-almost prime if it has at most R prime factors, counted with multiplicity. In this paper, we prove that there are infinitely many Beatty primes of the form ⌊ n^c ⌋ such that n is an R-almost primes with c ∈ (1, c_R) and c_R is an explicit constant depending on R.
We show that MacMahon’s sum-of-divisors q-series A_k(q) and C_k(q) arise naturally as the coefficients in the expansions of Gosper’s q-trigonometric functions. In particular, we express sin _q(π z) and cos _q(π z) in terms of MacMahon’s functions and use Gosper’s q-trigonometric identities to derive new convolution formulas for A_k(q) and C_k(q) . As an application, we obtain well-known modular identities relating the Eisenstein series E_2 and certain η -products.
