calling r ∈ Z+ the balancer corresponding to the balancing number n. The numbers 6, 35, and 204 are examples of balancing numbers with balancers 2, 14, and 84, respectively. Behera and Panda [1] also proved that a positive integer n is a balancing number if and only if n2 is a triangular number, that is, 8n2 + 1 is a perfect square. Though the definition of balancing numbers suggests that no ba...

MATILDE LALÍN, FRANCIS RODRIGUE, AND MATHEW ROGERS Abstract. We study the series ψs(z) := ∑∞ n=1 sec(nπz)n −s, and prove that it converges under mild restrictions on z and s. The function possesses a modular transformation property, which allows us to evaluate ψs(z) explicitly at certain quadratic irrational values of z. This supports our conjecture that πψk( √ j) ∈ Q whenever k and j are posit...

In the paper, the authors generalize the notion “k-Mittag-Leffler function”, establish some integral transforms of the generalized k-Mittag-Leffler function, and derive several special and known conclusions in terms of the generalized Wright function and the generalized k-Wright function. 1. Preliminaries Throughout this paper, let C, R, R0 , R, Z − 0 , and N denote respectively the sets of com...

[03.1] For a bounded sequence of complex numbers c n , prove that ∞ n=0 c n z n z n + 1 converges to a holomorphic function on |z| < 1. Each summand is holomorphic on |z| < 1, because of the quotient rule, and that the numerator and denominator are polynomials, hence holomorphic. To prove that the sum n f n of a sequence of holomorphic functions on |z| < 1 is itself holomorphic, it suffices to ...

Example 1.2. (a) Suppose X is a Riemann surface. Let Y ⊂ X be a (connected) open subset. Then Y is a Riemann surface, whose complex structure is given by taking all U ⊂ Y from charts of X. (b) Let P = C ∪ {∞}, homeomorphic to the real sphere. Take U1 = P\{∞} = C, U2 = P\{0} = C∗ ∪ {∞}. Define φ1(z) = z, φ2(z) = 1/z for z 6= ∞ and φ2(∞) = 0. Then φ2 ◦ φ−1 1 : C∗ → C∗ is given by z 7→ 1/z, which ...

Let Z and R be the sets of all integers and real numbers, respectively. For a,b ∈ Z, define Z(a)= {a,a+1, . . .} and Z(a,b)= {a,a+1, . . . ,b} when a≤ b. Let A be an n×m matrix. Aτ denotes the transpose ofA.When n=m, σ(A) and det(A) denote the set of eigenvalues and the determinant of A, respectively. In this paper, we study the existence of multiple p-periodic solutions to the following discre...

Let Z and R be the sets of all integers and real numbers, respectively. For a,b ∈ Z, define Z(a)= {a,a+1, . . .} and Z(a,b)= {a,a+1, . . . ,b} when a≤ b. Let A be an n×m matrix. Aτ denotes the transpose ofA.When n=m, σ(A) and det(A) denote the set of eigenvalues and the determinant of A, respectively. In this paper, we study the existence of multiple p-periodic solutions to the following discre...

The spectral properties of Lyman break galaxies (LBGs) offer a means to isolate pure samples displaying either dominant Lyα in absorption or Lyα in emission using broadband information alone. We present criteria developed using a large z ∼ 3 LBG spectroscopic sample from the literature that enables large numbers of each spectral type to be gathered in photometric data, providing good statistics...

We follow the rules: a, b, c, d, a′, b′, c′, d′, x, y, x1, u, v are real numbers and s, t, h, z, z1, z2, z3, z4, s1, s2, s3, p, q are elements of C. Let a be a real number and let us consider z. Then a · z is an element of C and it can be characterized by the condition: (Def. 1) a · z = (a + 0i) · z. Then a + z is an element of C and it can be characterized by the condition: (Def. 2) a + z = z ...

Unitary Matrices and Hermitian Matrices Recall that the conjugate of a complex number a + bi is a − bi. The conjugate of a + bi is denoted a+ bi or (a+ bi)∗. In this section, I’ll use ( ) for complex conjugation of numbers of matrices. I want to use ( )∗ to denote an operation on matrices, the conjugate transpose. Thus, 3 + 4i = 3− 4i, 5− 6i = 5 + 6i, 7i = −7i, 10 = 10. Complex conjugation sati...