scholarly journals Calculation of Augmented Jacobi Polynomials by Means of a Recurrence Relation

1980 ◽  
Vol 4 (2) ◽  
pp. 93-97 ◽  
Author(s):  
Zhi-de Liang ◽  
Jia-Zheng Xu ◽  
Fu Wang
Keyword(s):  
Recurrence Relation ◽  
Recurrence Relations ◽  
Jacobi Polynomials

A recurrence relation for Zℓmn(ξ) is deduced from the recurrence relations for Jacobi polynomials. Based on this relation an ALGOL-60 program has been written for efficiently calculating the Zℓmn(ξ) required in ODF analysis.

scholarly journals ELEMENTARY OSCILLATION CRITERIA FOR A THREE TERM RECURRENCE RELATION WITH OSCILLATORY COEFFICIENT SEQUENCE

1998 ◽  
Vol 29 (3) ◽  
pp. 227-232
Author(s):  
GUANG ZHANG ◽  
SUI-SUN CHENG
Keyword(s):  
Recurrence Relation ◽  
Recurrence Relations ◽  
Qualitative Properties ◽  
Oscillation Criteria ◽  
Three Term Recurrence Relation

Qualitative properties of recurrence relations with coefficients taking on both positive and negative values are difficult to obtain since mathematical tools are scarce. In this note we start from scratch and obtain a number of oscillation criteria for one such relation : $x_{n+1}-x_n+p_nx_{n-r}\le 0$.

scholarly journals On the quotient moment of lower generalized order statistics and characterization

2015 ◽  
Vol 11 (1) ◽  
pp. 73-89
Author(s):  
Devendra Kumar
Keyword(s):  
Order Statistics ◽  
Recurrence Relation ◽  
Conditional Expectation ◽  
General Class ◽  
Recurrence Relations ◽  
Generalized Order Statistics ◽  
Lower Records ◽  
Moments Of Order Statistics ◽  
Generalized Order

Abstract In this paper we consider general class of distribution. Recurrence relations satisfied by the quotient moments and conditional quotient moments of lower generalized order statistics for a general class of distribution are derived. Further the results are deduced for quotient moments of order statistics and lower records and characterization of this distribution by considering the recurrence relation of conditional expectation for general class of distribution satisfied by the quotient moment of the lower generalized order statistics.

scholarly journals Distinct partitions and overpartitions

2021 ◽  
Vol 38 (1) ◽  
pp. 149-158
Author(s):  
MIRCEA MERCA ◽  
Keyword(s):  
Partition Function ◽  
Recurrence Relation ◽  
Recurrence Relations ◽  
Convergent Series ◽  
The Other ◽  
Large Integer ◽  
Linear Recurrence ◽  
Fast Computation ◽  
Real Numbers ◽  
Very High

In 1963, Peter Hagis, Jr. provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the partition function $Q(n)$ which counts partitions of $n$ into distinct parts. Computing $Q(n)$ by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we investigate new connections between partitions into distinct parts and overpartitions and obtain a surprising recurrence relation for the number of partitions of $n$ into distinct parts. By particularization of this relation, we derive two different linear recurrence relations for the partition function $Q(n)$. One of them involves the thrice square numbers and the other involves the generalized octagonal numbers. The recurrence relation involving the thrice square numbers provide a simple and fast computation of the value of $Q(n)$. This method uses only (large) integer arithmetic and it is simpler to program. Infinite families of linear inequalities involving partitions into distinct parts and overpartitions are introduced in this context.

scholarly journals Some Recurrence Relations and Hilbert Series of Right-Angled Affine Artin Monoid M(D~n∞)

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Young Chel Kuwn ◽  
Zaffar Iqbal ◽  
Abdul Rauf Nizami ◽  
Mobeen Munir ◽  
Sana Riaz ◽  
...  
Keyword(s):  
Growth Rate ◽  
Recurrence Relation ◽  
Recurrence Relations ◽  
Hilbert Series ◽  
The Right

We find the Hilbert series of the right-angled affine Artin monoid M(D~n∞). We also discuss its recurrence relation and the growth rate.

The geometric unfolding of recurrence relations

2020 ◽  
Vol 104 (561) ◽  
pp. 403-411
Author(s):  
Stan Dolan
Keyword(s):  
Recurrence Relation ◽  
Recurrence Relations ◽  
Initial Values ◽  
Almost All

In 1942 R. C. Lyness challenged readers of the Gazette to find a recurrence relation of order 2 which would generate a cycle of period 7 for almost all initial values [1].

Explicit Formulas for the Associated Jacobi Polynomials and Some Applications

1987 ◽  
Vol 39 (4) ◽  
pp. 983-1000 ◽  
Author(s):  
Jet Wimp
Keyword(s):  
Weight Function ◽  
Closed Form ◽  
Recurrence Relation ◽  
Hypergeometric Functions ◽  
Continued Fraction ◽  
Jacobi Polynomials ◽  
Main Diagonal ◽  
Discrete Point ◽  
Explicit Formulas ◽  
Explicit Representations

In this paper we determine closed-form expressions for the associated Jacobi polynomials, i.e., the polynomials satisfying the recurrence relation for Jacobi polynomials with n replaced by n + c, for arbitrary real c ≧ 0. One expression allows us to give in closed form the [n — 1/n] Padé approximant for what is essentially Gauss' continued fraction, thus completing the theory of explicit representations of main diagonal and off-diagonal Padé approximants to the ratio of two Gaussian hypergeometric functions and their confluent forms, an effort begun in [2] and [19]. (We actually give only the [n — 1/n] Padé element, although other cases are easily constructed, see [19] for details.)We also determine the weight function for the polynomials in certain cases where there are no discrete point masses. Concerning a weight function for these polynomials, so many writers have obtained so many partial results that our formula should be considered an epitome rather than a real discovery, see the discussion in Section 3.

scholarly journals New Tribonacci recurrence relations and addition formulas

2020 ◽  
Vol 26 (4) ◽  
pp. 164-172
Author(s):  
Kunle Adegoke ◽  
◽  
Adenike Olatinwo ◽  
Winning Oyekanmi ◽  
◽  
...  
Keyword(s):  
Recurrence Relation ◽  
Recurrence Relations ◽  
Simple Relation ◽  
Addition Formula ◽  
Lucas Numbers ◽  
Tribonacci Numbers ◽  
Three Term Recurrence Relation ◽  
Special Case ◽  
Addition Formulas

Only one three-term recurrence relation, namely, W_{r}=2W_{r-1}-W_{r-4}, is known for the generalized Tribonacci numbers, W_r, r\in Z, defined by W_{r}=W_{r-1}+W_{r-2}+W_{r-3} and W_{-r}=W_{-r+3}-W_{-r+2}-W_{-r+1}, where W_0, W_1 and W_2 are given, arbitrary integers, not all zero. Also, only one four-term addition formula is known for these numbers, which is W_{r + s} = T_{s - 1} W_{r - 1} + (T_{s - 1} + T_{s-2} )W_r + T_s W_{r + 1}, where ({T_r})_{r\in Z} is the Tribonacci sequence, a special case of the generalized Tribonacci sequence, with W_0 = T_0 = 0 and W_1 = W_2 = T_1 = T_2 = 1. In this paper we discover three new three-term recurrence relations and two identities from which a plethora of new addition formulas for the generalized Tribonacci numbers may be discovered. We obtain a simple relation connecting the Tribonacci numbers and the Tribonacci–Lucas numbers. Finally, we derive quadratic and cubic recurrence relations for the generalized Tribonacci numbers.

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 Some identities and recurrence relations on the two variables Bernoulli, Euler and Genocchi polynomials

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1757-1765
Author(s):  
Veli Kurt ◽  
Burak Kurt
Keyword(s):  
Recurrence Relation ◽  
Recurrence Relations ◽  
Bernoulli Polynomials ◽  
Addition Theorem ◽  
Euler Polynomials ◽  
Genocchi Polynomials

Mahmudov in ([16], [17], [18]) introduced and investigated some q-extensions of the q-Bernoulli polynomials B(?)n,q (x,y) of order ?, the q-Euler polynomials ?(?)n,q (x,y) of order ? and the q-Genocchi polynomials G(?)n,q (x,y) of order ?. In this article, we give some identities for the q-Bernoulli polynomials, q-Euler polynomials and q-Genocchi polynomials and the recurrence relation between these polynomials. We give a different form of the analogue of the Srivastava-Pint?r addition theorem.

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