A new member of the Pell sequences: the pseudo-Pell sequence

In this study, we define a new type of Pell and Pell-Lucas numbers which are called Gaussian-bihyperbolic Pell and Pell-Lucas numbers. We also define negaGaussian-bihyperbolic Pell and Pell-Lucas numbers. Moreover, we obtain Binet’s formulas, generating function formulas, d’Ocagne’s identities, Catalan’s identities, Cassini’s identities and some sum formulas for these new type numbers and we investigate some algebraic proporties of these. Furthermore, we give the matrix representation of Gaussian-bihyperbolic Pell and Pell-Lucas numbers.

In a recent note, Santana and Diaz-Barrero proved a number of sum identities involving the well-known Pell numbers. Their proofs relied heavily on the Binet formula for the Pell numbers. Our goal in this note is to reconsider these identities from a purely combinatorial viewpoint. We provide bijective proofs for each of the results by interpreting the Pell numbers as enumerators of certain types of tilings. In turn, our proofs provide helpful insight for straightforward generalizations of a number of the identities.

The k-Fibonacci sequence starts with the values (a total of k terms) and each term afterwards is the sum of the k preceding terms. In this paper, we find all integers c having at least two representations as a difference between a k-Fibonacci number and a Pell number. This paper continues and extends the previous work of [J.J. Bravo, C.A. Gómez, and J.L. Herrera, On the intersection of k-Fibonacci and Pell numbers, Bull. Korean Math. Soc. 56(2) (2019), pp. 535–547; S. Hernández, F. Luca, and L.M. Rivera, On Pillai's problem with the Fibonacci and Pell sequences, Soc. Mat. Mex. 25 (2019), pp. 495–507 and M.O. Hernane, F. Luca, S.E. Rihane, and A. Togbé, On Pillai's problem with Pell numbers and powers of 2, Hardy- Ramanujan J. 41 (2018), pp. 22–31].

We obtain the Binet’s formula for k-Pell numbers and as a consequence we get some properties for k-Pell numbers. Also we give the generating function for k-Pell sequences and another expression for the general term of the sequence, using the ordinary generating function, is provided. Mathematics Subject Classification: 11B37, 05A15, 11B83.

In this study, we investigate Pell numbers and Leonardo numbers and describe a new third-order number sequence entitled Pell Leonardo numbers. We then construct some identities, including the Binet formula, generating function, exponential generating function, Catalan, Cassini, and d’Ocagne’s identities for Pell Leonardo numbers and obtain a relation between Pell Leonardo and Pell numbers. In addition, we present some summation formulas of Pell Leonardo numbers based on Pell numbers. Finally, we create a matrix formula for Pell Leonardo numbers and obtain the determinant of the matrix.

In this paper we give a new generalization of the Pell numbers in matrix representation. Also we extend the matrix representation and we show that the sums of the generalized order-$k$ Pell numbers could be derived directly using this representation. Further we present some identities, the generalized Binet formula and combinatorial representation of the generalized order-$k$ Pell numbers.

In this paper, we compute terms of the matrix $A_{(k)}^{\infty }$, which contains Fibonacci type numbers and polynomials, with the help of determinants and permanents of various Hessenberg matrices. In addition, we show that determinants of these Hessenberg matrices can be obtained by using combinations. The results that we obtain are important, since the matrix $A_{(k)}^{\infty } $ is a general form of Fibonacci type numbers and polynomials, such as $k$ sequences of the generalized order-$k$ Fibonacci and Pell numbers, generalized bivariate Fibonacci $p$-polynomials, bivariate Fibonacci and Pell $p$-polynomials, second kind Chebyshev polynomials and bivariate Jacobsthal polynomials, etc.

In this paper, we introduce and study a new two-parameters generalization of the Fibonacci numbers, which generalizes Fibonacci numbers, Pell numbers, and Narayana numbers, simultaneously. We prove some identities which generalize well-known relations for Fibonacci numbers, Pell numbers and their generalizations. A matrix representation for generalized Fibonacci numbers is given, too.

In this study, using the Leonardo numbers, we define a new type of quaternion that is called a Leonard Gaussian quaternion. We also give a negative-Leonardo Gaussian quaternion. These numbers are introduced from the set of complex numbers and quaternions. Moreover, we obtain the Binet’s formula, generating function formula, d’Ocagne’s identity, Catalan’s identity, Cassini’s identity, Honsberger’s identity, like-Vajda’s identity and some formulas for these new types of numbers. Morever, we give the matrix representation of the Leonardo Gaussian quaternion.

In the number theory additive problems is very important. One of them is the Ingam binary additive divisor problem on the representation of natural number as the difference of product of numbers. Many mathematician like T. Esterman, D. I. Ismoilov, D. R. Heath-Brown, G. I. Arkhipov and V. N. Chubarikov, J.-M. Deshouillers and H. Iwaniec improved the remainder term in the asymptotic formula of the number of solution of this diophantine equation. In present paper one problem with quadratic forms is considered. This problem is analog of the Ingam binary additive divisor problem. Let d — negative square-free number, F = Q( √ d) — imaginary quadratic field, δF — discriminant of field F, Q1(m), Q2(k) — binary positive defined primitive quadratic forms with matrixes A1, A2, det A1 = det A2 = −δF , e > 0 — arbitrarily small number; n ∈ N, h ∈ N. The asymptotical formula of the number of solution of diophantine equation Q1(m) − Q2(k) = h with weight coefficient exp ( −(Q1(m) + Q2(k))/n) is received. In this asymptotical formula discriminant of field δF is fixed and the remainder term is estimating as O(h en 3/4+e ), which not depend of δF . Moreover the parameter h grow as O(n) with growing on the main parameter n. Proof of the asymptotical formula based on circular method when sum, which is solution of diophantine equation, may be representing as integral. Interval of integration divided by numbers of Farey series. The taking weight coefficient allow to use the functional equation of the theta-function. Moreover the estimation of one sum with Gauss sums is important. Using the evident formula of some product of Gauss sums of the number which coprimes of discriminant of field this sum represented of Kloosterman’s sum which estimate by A. Weil.

Double Hurwitz numbers enumerate branched covers of {{{mathbb {C}}}}{{{mathbb {P}}}}^1 with prescribed ramification over two points and simple ramification elsewhere. In contrast to the single case, their underlying geometry is not well understood. In previous work by the second- and third-named authors, the double Hurwitz numbers were conjectured to satisfy a polynomiality structure and to be governed by the topological recursion, analogous to existing results concerning single Hurwitz numbers. In this paper, we resolve these conjectures by a careful analysis of the semi-infinite wedge representation for double Hurwitz numbers. We prove an ELSV-like formula for double Hurwitz numbers, by deforming the Johnson–Pandharipande–Tseng formula for orbifold Hurwitz numbers and using properties of the topological recursion under variation of spectral curves. In the course of this analysis, we unveil certain vanishing properties of Omega -classes.

In this paper we introduce a new kind of generalized Jacobsthal numbers in a distance sense. We give the identities and matrix representations for them and their connections with the Fibonacci and the Pell numbers. We also describe the interpretations of these numbers in terms of some kind of (k 1 A 1, k 2 A 2, k 3 A 3)-edge colouring and quasi colouring.

In this paper, we define the Hadamard-type k-step Pell sequence by using the Hadamard-type product of characteristic polynomials of the Pell sequence and the k-step Pell sequence. Also, we derive the generating matrices for these sequences, and then we obtain relationships between the Hadamard-type k-step Pell sequences and these generating matrices. Furthermore, we produce the Binet formula for the Hadamard-type k-step Pell numbers for the case that k is odd integers and k ≥ 3. Finally, we derive some properties of the Hadamard-type k-step Pell sequences such as the combinatorial representation, the generating function, and the exponential representation by using its generating matrix.

In this study, we define a new type of Pell and Pell-Lucas numbers which are called biGaussian Pell and biGaussian Pell-Lucas numbers. We also give the relationship between negabiGaussian Pell and Pell-Lucas numbers and bicomplex Pell and Pell-Lucas numbers. Moreover, we obtain the Binet’s formula, generating function, d’Ocagne’s identity, Catalan’s identity, Cassini’s identity and some sums formulas for these new type numbers. Some algebraic proporties of biGaussian Pell and Pell-Lucas numbers which are connected between biGaussian numbers and Pell and Pell-Lucas numbers are investigated. Moreover, we give the matrix representation of biGaussian Pell and Pell-Lucas numbers.

In this study, we define a new type of Pell and Pell-Lucas numbers which are called dual-Gaussian Pell and dual-Gaussian Pell-Lucas numbers. We also give the relationship between negadual-Gaussian Pell and Pell-Lucas numbers and dual-complex Pell and Pell-Lucas numbers. Also, some sum ve product properties of Pell and Pell-Lucas numbers are given. Moreover, we obtain the Binet’s formula, generating function, d’Ocagne’s identity, Catalan’s identity, Cassini’s identity and some sum formulas for these new type numbers. Some algebraic proporties of dual-Gaussian Pell and Pell-Lucas numbers are investigated. Futhermore, we give the matrix representation of dual-Gaussian Pell and Pell-Lucas numbers.