scholarly journals The Explicit Formula of the Presumed Optimal Recurrence Relation for the Star Tower of Hanoi

2019 ◽  
Vol E102.D (3) ◽  
pp. 492-498
Author(s):  
Akihiro MATSUURA ◽  
Yoshiaki SHOJI
Keyword(s):  
Recurrence Relation ◽  
Explicit Formula ◽  
Tower Of Hanoi

scholarly journals On the Star Puzzle

2019 ◽  
Vol 39 ◽  
pp. 1-14
Author(s):  
AAK Majumdar
Keyword(s):  
Recurrence Relation ◽  
Additional Condition ◽  
Tower Of Hanoi ◽  
Minimum Number

In the star puzzle, there are four pegs, the usual three pegs, S, P and D, and a fourth one at 0. Starting with a tower of n discs on the peg P, the objective is to transfer it to the peg D, in minimum number of moves, under the conditions of the classical Tower of Hanoi problem and the additional condition that all disc movements are either to or from the fourth peg. Denoting by MS(n) the minimum number of moves required to solve this variant, MS(n) satisfies the recurrence relation . This paper studies rigorously and extensively the above recurrence relation, and gives a solution of it. GANIT J. Bangladesh Math. Soc.Vol. 39 (2019) 1-14

scholarly journals AN INNOVATIVE STUDY ON SUM OF POWERS OF INTEGERS

2021 ◽  
Vol 12 (4) ◽  
pp. 1260-1266
Author(s):  
B.Mahaboob, Et. al.
Keyword(s):  
Number Theory ◽  
Recurrence Relation ◽  
Explicit Formula ◽  
Linear Algebra ◽  
Interesting Problem ◽  
Natural Numbers ◽  
Research Article ◽  
Unique Polynomial

The generalization of sum of integral powers of first n-natural numbers has been an interesting problem among the researchers in Analytical Number Theory for decades. This research article mainly focuses on the derivation of generalized result of this sum. More explicit formula has been derived in order to get the sum of any arbitrary integral powers of first n-natural numbers. Furthermore by using the fundamental principles of Combinatorics and Linear Algebra an attempt has been made to answer an interesting question namely: Is the sum of integral powers of natural numbers a unique polynomial? As a result it is depicted that this sum always equals a unique polynomial over natural numbers. Moreover some properties of the coefficients of this polynomial are derived.More importantly a recurrence relation which can give the formulas for sum of any positive integral powers of first n-natural numbers has been proposed and it is strongly believed that this recurrence relation is the most significant thing in this entire discussion

scholarly journals Explicit Formulas for the First Form (q,r)-Dowling Numbers and (q,r)-Whitney-Lah Numbers

2021 ◽  
Vol 14 (1) ◽  
pp. 65-81
Author(s):  
Roberto Bagsarsa Corcino ◽  
Jay Ontolan ◽  
Maria Rowena Lobrigas
Keyword(s):  
Recurrence Relation ◽  
Explicit Formula ◽  
Taylor Series ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Difference Operator ◽  
Interpolation Formula ◽  
Explicit Formulas ◽  
Fundamental Properties ◽  
Whitney Numbers

In this paper, a q-analogue of r-Whitney-Lah numbers, also known as (q,r)-Whitney-Lah number, denoted by $L_{m,r} [n, k]_q$ is defined using the triangular recurrence relation. Several fundamental properties for the q-analogue are established such as vertical and horizontal recurrence relations, horizontal and exponential generating functions. Moreover, an explicit formula for (q, r)-Whitney-Lah number is derived using the concept of q-difference operator, particularly, the q-analogue of Newton’s Interpolation Formula (the umbral version of Taylor series). Furthermore, an explicit formula for the first form (q, r)-Dowling numbers is obtained which is expressed in terms of (q,r)-Whitney-Lah numbers and (q,r)-Whitney numbers of the second kind.

scholarly journals ON A GENERALIZATION OF CATALAN'S POLYNOMIALS

2018 ◽  
Vol 33 (2) ◽  
pp. 163
Author(s):  
Goubi Mouloud
Keyword(s):  
Recurrence Relation ◽  
Explicit Formula ◽  
Special Class ◽  
Combinatorial Identities ◽  
New Class

Abstract. In this work, we define and study the generalized class of Catalan’s polynomials.Thereafter we connect them to the class of Humbert’s polynomials and re-foundthe Humbert recurrence relation [5]. This idea helps us to define a new class of generalizedHumbert’s polynomials different of those given by H. W. Gould [4] and P. N.Shrivastava [9]. Finally we establish an explicit formula for a special class of generalizedCatalan’s polynomials and get two useful combinatorial identities.

BRST OPERATOR FOR QUANTUM LIE ALGEBRAS: EXPLICIT FORMULA

2004 ◽  
Vol 19 (supp02) ◽  
pp. 240-247 ◽  
Author(s):  
A. P. ISAEV ◽  
O. OGIEVETSKY
Keyword(s):  
Quantum Group ◽  
Recurrence Relation ◽  
Explicit Formula ◽  
Lie Algebras ◽  
Important Class ◽  
Quadratic Algebras ◽  
Brst Operator ◽  
Quantum Lie Algebras

We continue our study of quantum Lie algebras, an important class of quadratic algebras arising in the Woronowicz calculus on a quantum group. Quantum Lie algebras are generalizations of Lie (super)algebras. Many notions from the theory of Lie (super)algebras admit "quantum" analogues. In particular, there is a BRST operator Q(Q2=0) which generates the differential in the Woronowicz theory and gives information about (co)homologies of quantum Lie algebras. In our previous papers a recurrence relation for the operator Q for quantum Lie algebras was given. Here we solve this recurrence relation and obtain an explicit formula for the BRST operator.

scholarly journals On a Recurrence Relation Related to The Reve’s Puzzle

2018 ◽  
Vol 42 (2) ◽  
pp. 191-199
Author(s):  
AAK Majumdar
Keyword(s):  
Recurrence Relation ◽  
Value Function ◽  
Optimal Value Function ◽  
Optimality Equation ◽  
Tower Of Hanoi ◽  
Optimal Value ◽  
Academy Of Sciences ◽  
Local Value

The 4-peg Tower of Hanoi problem, commonly known as the Reve’s puzzle, is well-known. Motivated by the optimality equation satisfied by the optimal value function M(n) satisfied in case of the Reve’s puzzle, (Matsuura et al. 2008) posed the following generalized recurrence relation T(n, a) = min {aT(n-t, a)+S(t,3)}             1≤ t ≤ n where n ≥ 1 and a ≥ 2 are integers, and S(t, 3) = 2t – 1 is the solution of the 3-peg Tower of Hanoi problem with t discs. Some local-value relationships are satisfied by T(n, a) (Majumdar et al. 2016). This paper studies the properties of  T(n+1, a) – T(n, a) more closely for the case when a is an integer not of the form 2i for any integer i ≥ 2. Journal of Bangladesh Academy of Sciences, Vol. 42, No. 2, 191-199, 2018

scholarly journals Enumeration of Hamiltonian cycles on a thick grid cylinder - part I: Non-contractible Hamiltonian cycles

2019 ◽  
Vol 13 (1) ◽  
pp. 28-60 ◽  
Author(s):  
Olga Bodroza-Pantic ◽  
Harris Kwong ◽  
Rade Doroslovacki ◽  
Milan Pantic
Keyword(s):  
Recurrence Relation ◽  
Explicit Formula ◽  
Positive Characteristic ◽  
Characteristic Root ◽  
Hamiltonian Cycles ◽  
Computational Data

In a recent paper, we have studied the enumeration of Hamiltonian cycles (abbreviated HCs) on the grid cylinder graph Pm+1 x Cn, where m grows while n is fixed. In this sequel, we study a much harder problem of enumerating HCs on the same graph only this time letting n grow while m is fixed. We propose a characterization for non-contractible HCs which enables us to prove that their numbers hnc, m (n) satisfy a recurrence relation for every fixed m. From the computational data, we conjecture that the coefficient for the dominant positive characteristic root in the explicit formula for hnc,m (n) is 1.

scholarly journals Asymptotic expansions for certain mathematical constants and special functions

2018 ◽  
Vol 12 (2) ◽  
pp. 493-507 ◽  
Author(s):  
Chao-Ping Chen
Keyword(s):  
Recurrence Relation ◽  
Explicit Formula ◽  
Asymptotic Expansions ◽  
Special Functions ◽  
Stirling Numbers ◽  
Mathematical Constants

For fixed real b > 1 and ? > 0, let S[?]b (n) = ?n,k=1 bkk-?. Abel proved that S[?]b(n) ~ bn ??,k =0 ckn-(k+?)(n ? ?), and gave an explicit formula for determining the coefficients ck ? ck(b,?) in terms of Stirling numbers of the second kind. We here provide a recurrence relation for determining the coefficients ck, without Stirling numbers. We also consider asymptotic expansions concerning Somos' quadratic recurrence constant, Glaisher-Kinkelin constant, Choi-Srivastava constants, and the Barnes G-function.

scholarly journals A Note on the (p, q)-Hermite Polynomials

2017 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz ◽  
Ayhan Esi ◽  
Serkan Araci
Keyword(s):  
Integral Representation ◽  
Recurrence Relation ◽  
Explicit Formula ◽  
Generating Function ◽  
Bernstein Polynomials ◽  
Hermite Polynomials ◽  
Exponential Generating Function

In this paper, we introduce a new generalization of the Hermite polynomials via (p, q)-exponential generating function and investigate several properties and relations for mentioned polynomials including derivative property, explicit formula, recurrence relation, integral representation. We also de…ne a (p, q)-analogue of the Bernstein polynomials and acquire their some formulas. We then provide some (p, q)-hyperbolic representations of the (p, q)-Bernstein polynomials. In addition, we obtain a correlation between (p, q)-Hermite polynomials and (p, q)-Bernstein polynomials.

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