The (p,q)-sine and (p,q)-cosine polynomials and their associated (p,q)-polynomials (2024)

Table of Contents

Abstract References FAQs References

Abstract

The introduction of two-parameter $(p, q)$ -calculus and Lie algebras in 1991has spurred a wave of recent research into $(p, q)$ -special polynomials, including $(p, q)$ -Bernoulli, $(p, q)$ -Euler, $(p, q)$ -Genocchi and $(p, q)$ -Frobenius–Euler polynomials.These investigations have been carried out by numerous researchers in order to uncover a wide range of identities associated with these polynomials and applications. In this article, we aim to introduce $(p, q)$ -sine and $(p, q)$ -cosine based λ-array type polynomials and derive numerous properties of these polynomialssuch as $(p, q)$ -integral representations, $(p, q)$ -partial derivative formulae and $(p, q)$ -addition formulae.It is worth noting that the utilization of the $(p, q)$ -polynomials introduced in this study, along with other $(p, q)$ -polynomials, can lead to the derivation of various identities that differ from the ones presented here.

Keywords: Stirling number of second kind

MSC 2020: 05A30; 11B68; 11B73; 11B15; 11B83; 33D15

Funding statement: The third author Talha Usman would like to thank Scientific Research Department at University of Technology and Applied Sciences, Sur for supporting this work under Project No. UTAS-Sur-SRD-IRF 23-04/06.

References

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See Also

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Received: 2023-05-12

Revised: 2023-08-21

Accepted: 2023-09-03

Published Online: 2023-11-30

Published in Print: 2024-02-01

The (p,q)-sine and (p,q)-cosine polynomials and their associated
(p,q)-polynomials (2024)

FAQs

What is P and Q in a polynomial function? ›

The rational root theorem says, a rational zero of a polynomial is of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

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What is the name for the quotient of two polynomials p and q where q is not zero? ›

Any rational expression can be written as the quotient of two polynomials P/Q with Q ≠ 0, although this representation isn't unique.

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Is P and Q are two polynomials of degree 5 and 4? ›

It has been given that P and Q are two polynomials of degree 5 and 4 respectively. So, the highest exponent of P is 5 and that of Q is 4. When we subtract these two polynomials then the highest exponent of the resultant polynomial will also be 5. Therefore, the degree of p-q will be 5.

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Are sin and cos polynomials? ›

The trigonometric functions are periodic and cannot be described by a polynomial.

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What is P by q formula? ›

Answer: In mathematics, a rational number is a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. For example, −3/7 is a rational number, as is every integer (e.g. 5 = 5/1).

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How to find p and q? ›

You can calculate the p and q by using the total number of alleles of p or q divided by the total number of alleles in the population or finding q^2 to find q.

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What is a 4 degree polynomial called? ›

A bi-quadratic polynomial. A polynomial of degree 4 is called bi-quadratic polynomial.

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What are two polynomials with 4 or more terms? ›

[3xyz - 78p^2 + 11xy + 923y^5 - 56p + 41] and [89ab - 5abc + 13c^2 - 37a^3 + 21b] are two polynomials with four or more terms.

What are the 5 kinds of polynomial equation according to degree? ›

Types of Polynomials Based on its Degree

Degree	Polynomial Name
Degree 1	Linear Polynomial
Degree 2	Quadratic Polynomial
Degree 3	Cubic Polynomial
Degree 4	Quartic Polynomial

1 more row

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What is the relationship between sine and cosine? ›

sin(x degrees) = cos(90 - x degrees), and vice versa.

What is the sine and cosine rule? ›

The sine rule can be used to find an angle from 3 sides and an angle, or a side from 3 angles and a side. The cosine rule can find a side from 2 sides and the included angle, or an angle from 3 sides.

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Are sin and cos always equal? ›

So the only point at which sine and cosine are equal is where interchanging the adjacent and opposite sides makes no difference - that is, when they're equal to each other, which happens only in the case of an isosceles triangle.

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What does P mean in polynomials? ›

A polynomial is generally represented as P(x). The highest power of the variable of P(x) is known as its degree. Degree of a polynomial function is very important as it tells us about the behaviour of the function P(x) when x becomes very large. The domain of a polynomial function is entire real numbers (R).

What is P and q in quadratic functions? ›

'p' and 'q' represent the x-intercepts. The x-intercepts are the points where the graph crosses the x-axis.

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What is P and q in a statement? ›

In conditional statements, "If p then q" is denoted symbolically by "p q"; p is called the hypothesis and q is called the conclusion. For instance, consider the two following statements: If Sally passes the exam, then she will get the job.

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What does P and q stand for in algebra? ›

The P stands for the Predicate, and the Q stands for the Quantifier. This is similar in math to how X is often the independent variable, and Y is often the dependent variable in an equation.

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