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

[1] A. Belafhal, H. Benzehoua, A. Balhamri and T. Usman,An advanced method for evaluating Lommel integral and its application in marine environment,J. Comput. Appl. Math. 416 (2022), Paper No. 114600.10.1016/j.cam.2022.114600Search in Google Scholar

[2] A. Belafhal, H. Benzehoua and T. Usman,Certain integral transforms and their application to generate new Laser waves: Exton–Gaussian beams,Adv. Math. Models Appl. 6 (2021), no. 3, 206–217.Search in Google Scholar

[3] A. Belafhal, S. Chib, F. Khannous and T. Usman,Evaluation of integral transforms using special functions with applications to biological tissues,Comput. Appl. Math. 40 (2021), no. 4, Paper No. 156.10.1007/s40314-021-01542-2Search in Google Scholar

See Also
Representation of $(p,q)$ -Bernstein polynomials in terms of $(p,q)$ -Jacobi polynomialsRational Root Theorem (Rational Zero Theorem) - Examples, Proof

[4] A. Belafhal, E. M. El Halba and T. Usman,An integral transform involving the product of Bessel functions and Whittaker function and its application,Int. J. Appl. Comput. Math. 6 (2020), no. 6, Paper No. 177.10.1007/s40819-020-00930-2Search in Google Scholar

[5] A. Belafhal, E. M. El Halba and T. Usman,A note on some representations of Appell and Horn functions,Adv. Stud. Contemp. Math. (Kyungshang) 30 (2020), 5–16.Search in Google Scholar

[6] A. Belafhal, E. M. El Halba and T. Usman,Certain integral transforms and their applications in propagation of Laguerre–Gaussian Schell-model beams,Commun. Math. 29 (2021), no. 3, 483–491.10.2478/cm-2021-0030Search in Google Scholar

[7] A. Belafhal, E. M. El Halba and T. Usman,An integral transform and its application in the propagation of Lorentz–Gaussian beams,Contemp. Math. to appear.Search in Google Scholar

[8] A. Belafhal and S. Hennani,A note on some integrals used in laser field involving the product of Bessel functions,Phys. Chem. News 61 (2011), 59–62.Search in Google Scholar

[9] A. Belafhal, N. Nossir, L. Dalil-Essakali and T. Usman,Integral transforms involving the product of Humbert and Bessel functions and its application,AIMS Math. 5 (2020), no. 2, 1260–1274.10.3934/math.2020086Search in Google Scholar

[10] N. P. Cakić and G. V. Milovanović,On generalized Stirling numbers and polynomials,Math. Balkanica (N. S.) 18 (2004), no. 3–4, 241–248.Search in Google Scholar

[11] R. Chakrabarti and R. Jagannathan,A ( p , q ) -oscillator realization of two-parameter quantum algebras,J. Phys. A 24 (1991), no. 13, L711–L718.10.1088/0305-4470/24/13/002Search in Google Scholar

[12] L. Comtet,Advanced Combinatorics. The Art of Finite and Infinite Expansions,D. Reidel, Dordrecht, 1974.10.1007/978-94-010-2196-8Search in Google Scholar

[13] R. B. Corcino,On p , q -binomial coefficients,Integers 8 (2008), Paper No. A29.Search in Google Scholar

[14] A. de Médicis and P. Leroux,Generalized Stirling numbers, convolution formulae and p , q -analogues,Canad. J. Math. 47 (1995), no. 3, 474–499.10.4153/CJM-1995-027-xSearch in Google Scholar

[15] U. Duran, M. Acikgoz and S. Araci,On ( p , q ) -Bernoulli, ( p , q ) -Euler and ( p , q ) -Genocchi polynomials,J. Comput. Theor. Nanosci. 13 (2016), no. 11, 7833–7846.10.1166/jctn.2016.5785Search in Google Scholar

[16] U. Duran and M. Acikgoz,Apostol type ( p , q ) -Frobenius–Euler polynomials and numbers,Kragujevac J. Math. 42 (2018), no. 4, 555–567.10.5937/KgJMath1804555DSearch in Google Scholar

[17] U. Duran, M. Acikgoz and S. Araci,On some polynomials derived from ( p , q ) -calculus,J. Comput. Theor. Nanosci. 13 (2016), no. 11, 7903–7908.10.1166/jctn.2016.5790Search in Google Scholar

[18] U. Duran, M. Acikgoz and S. Araci,A study on some new results arising from ( p , q ) -calculus,TWMS J. Pure Appl. Math. 11 (2020), no. 1, 57–71.Search in Google Scholar

[19] H. W. Gould,Combinatorial Identities: A Standardized Set of Tables Listing 500 Binomial Coefficient Summations,Henry W. Gould, Morgantown, 1972.Search in Google Scholar

[20] V. Kac and P. Cheung,Quantum Calculus,Universitext,Springer, New York, 2002.10.1007/978-1-4613-0071-7Search in Google Scholar

[21] N. Khan and S. Husain,Analysis of Bell based Euler polynomials and their application,Int. J. Appl. Comput. Math. 7 (2021), no. 5, Paper No. 195.10.1007/s40819-021-01127-xSearch in Google Scholar

[22] N. Khan and S. Husain,A family of Apostol–Euler polynomials associated with Bell polynomials,Analysis (Berlin) 43 (2023), 10.1515/anly-2022-1101.10.1515/anly-2022-1101Search in Google Scholar

[23] N. Khan, S. Husain, T. Usman and S. Araci,A new family of Apostol–Genocchi polynomials associated with their certain identities,Appl. Math. Sci. Eng. 31 (2023), no. 1, 1–19.10.1080/27690911.2022.2155641Search in Google Scholar

[24] W. A. Khan, G. Muhiuddin, U. Duran and D. Al-Kadi,On ( p , q ) -Sine and ( p , q ) -Cosine Fubini polynomials,Symmetry 14 (2022), no. 3, Article ID 527.10.3390/sym14030527Search in Google Scholar

[25] Q.-M. Luo and H. M. Srivastava,Some generalizations of the Apostol–Genocchi polynomials and the Stirling numbers of the second kind,Appl. Math. Comput. 217 (2011), no. 12, 5702–5728.10.1016/j.amc.2010.12.048Search in Google Scholar

[26] M. Masjed-Jamei and W. Koepf,Symbolic computation of some power-trigonometric series,J. Symbolic Comput. 80 (2017), 273–284.10.1016/j.jsc.2016.03.004Search in Google Scholar

[27] M. Masjed-Jamei, G. V. Milovanović and M. C. Dağlı,A generalization of the array type polynomials,Math. Morav. 26 (2022), no. 1, 37–46.10.5937/MatMor2201037MSearch in Google Scholar

[28] M. Masjed-Jamei and Z. Moalemi,Sine and cosine types of generating functions,Appl. Anal. Discrete Math. 15 (2021), no. 1, 82–105.10.2298/AADM200530002MSearch in Google Scholar

[29] P. Njionou Sadjang,On two ( p , q ) -analogues of the Laplace transform,J. Difference Equ. Appl. 23 (2017), no. 9, 1562–1583.Search in Google Scholar

[30] P. Njionou Sadjang,On the fundamental theorem of ( p , q ) -calculus and some ( p , q ) -Taylor formulas,Results Math. 73 (2018), no. 1, Paper No. 39.10.1007/s00025-018-0783-zSearch in Google Scholar

[31] P. Njionou Sadjang,On ( p , q ) -Appell polynomials,Anal. Math. 45 (2019), no. 3, 583–598.10.1007/s10476-019-0826-zSearch in Google Scholar

[32] P. Njionou Sadjang and U. Duran,On two bivariate kinds of ( p , q ) -Bernoulli polynomials,Miskolc Math. Notes 20 (2019), no. 2, 1185–1199.10.18514/MMN.2019.2587Search in Google Scholar

[33] J. Riordan,An Introduction to Combinatorial Analysis,John Wiley and Sons, New York, 1958.Search in Google Scholar

[34] Y. Simsek,Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications,Fixed Point Theory Appl. 2013 (2013), Paper No. 87.10.1186/1687-1812-2013-87Search in Google Scholar

[35] Y. F. Smirnov and R. F. Wehrhahn,The Clebsch–Gordan coefficients for the two-parameter quantum algebra SU p , q ⁢ ( 2 ) in the Löwdin–Shapiro approach,J. Phys. A 25 (1992), no. 21, 5563–5576.10.1088/0305-4470/25/21/015Search in Google Scholar

[36] H. M. Srivastava and J. Choi,Zeta and q-Zeta Functions and Associated Series and Integrals,Elsevier, Amsterdam, 2012.10.1016/B978-0-12-385218-2.00002-5Search in Google Scholar

[37] T. Usman, M. Saif and J. Choi,Certain identities associated with ( p , q ) -binomial coefficients and ( p , q ) -Stirling polynomials of the second kind,Symmetry 12 (2020), no. 9, Article ID 1436.10.3390/sym12091436Search in Google Scholar

[38] http://en.wikipedia.org/wiki/Bell_polynomials.Search in Google Scholar

[39] https://en.wikipedia.org/wiki/Fa\`{a}_di_Bruno's_formula.Search in Google Scholar

Received: 2023-05-12

Revised: 2023-08-21

Accepted: 2023-09-03

Published Online: 2023-11-30

Published in Print: 2024-02-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

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

FAQs

Are sin and cos polynomials? ›

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

Tell Me More ›
What is a polynomial function in math? ›

A polynomial function is a function that involves only non-negative integer powers or only positive integer exponents of a variable in an equation like the quadratic equation, cubic equation, etc. For example, 2x+5 is a polynomial that has exponent equal to 1.

Explore More ›
Is there a relationship between sin and cos? ›

Since we are talking about a right triangle, the angles are complementary. And this fact gives us enough information to conclude the following equation: sin(x degrees) = cos(90 - x degrees), and vice versa.

Learn More ›
What does cos and sin equal to? ›

Now as per sine, cosine and tangent formulas, we have here: Sine θ = Opposite side/Hypotenuse = BC/AC. Cos θ = Adjacent side/Hypotenuse = AB/AC. Tan θ = Opposite side/Adjacent side = BC/AB.

Continue Reading ›
What are 5 examples of polynomials? ›

Examples of polynomials:
  • 2a + 5b is a polynomial of two terms in two variables a and b.
  • 3xy + 5x + 1 is a polynomial of three terms in two variables x and y.
  • 3y4 + 2y3 + 7y2 – 9y + 3/5 is a polynomial of five terms in two variables x and y.
  • m + 5mn – 7m2n + nm2 + 9 is a polynomial of four terms in two variables m and n.

Learn More ›
How do you identify a polynomial? ›

The polynomials can be identified by noting which expressions contain only the operations of addition, subtraction, multiplication, and non-negative integer exponents. The non-polynomial expressions will be the expressions which contain other operations. Explain why the non-polynomial expressions are not polynomials.

View More ›
How to calculate a polynomial? ›

1.1: Solve Polynomial Equations by Factoring
  1. Difference of squares:a2−b2=(a−b)(a+b)
  2. Sum of squares: a2+b2 no general formula.
  3. Difference of cubes: a3−b3=(a−b)(a2+ab+b2)
  4. Sum of cubes: a3+b3=(a+b)(a2−ab+b2)
  5. If a binomial is both a difference of squares and a difference cubes, then first factor it as difference of squares.
Sep 13, 2022

Continue Reading ›
Is sine a polynomial? ›

A sine-polynomial p(x, sin x) is an analytic function which can be considered as a polynomial with real coefficients depending on the variables x and y=sin x. Let us remark that a sine-polynomial is defined on the real axis and the set of its roots is discrete.

Get More Info ›
Are trig functions considered polynomials? ›

Trigonometric functions

The function f(t) is not a rational function because the denominator is not a polynomial. On the other hand, the function g(y) can be rewritten as a fraction with numerator 3y and denominator (y+1)(y+1), both of which are polynomials, so it is a rational function. (Is the denominator obvious?

Find Out More ›
Why isn't Sinx a polynomial? ›

Odd power functions have odd symmetry and even power functions have even symmetry. So a polynomial comprised of only odd power functions still has odd symmetry and likewise for even. Since the cosine function has even symmetry its polynomial representation cannot have any odd powers and likewise for sine.

See More ›

Which polynomials are? ›

Degree of a Polynomial
PolynomialDegreeExample
Linear Polynomial1P(x) = 3x+1
Quadratic Polynomial2P(x) = 4x2+1x+1
Cubic Polynomial3P(x) = 6x3+4x2+3x+1
Quartic Polynomial4P(x) = 6x4+3x3+3x2+2x+1
2 more rows

See Details ›

References

Top Articles
13 Fascinating Facts About Black Cats
Ten Facts about the Red Panda That Makes It Special - Google Arts & Culture
Miami, FL Rooms for Rent - Short & Long-Term Rentals | iROOMit
Afton Hockey
Metra Union Pacific West Schedule
Add To List If Not Null C#
10-Day Weather Forecast for Vienna, WV - The Weather Channel | weather.com
'The Greatest Social Media Site Is Craigslist' - Slashdot
Foglesong Funeral Home : Mason, West Virginia (WV)
Our Staff | Foglesong Funeral Home
19660 W Doe Drive, Georgetown, DE 19947 For Sale
7 Frankenberry Drive, Georgetown, DE 19947 | Compass
Latest Posts
16 Remarkable Red Panda Facts - Fact Animal
Red Panda - Description, Habitat, Image, Diet, and Interesting Facts
Article information

Author: Dong Thiel

Last Updated:

Views: 6347

Rating: 4.9 / 5 (79 voted)

Reviews: 94% of readers found this page helpful

Author information

Name: Dong Thiel

Birthday: 2001-07-14

Address: 2865 Kasha Unions, West Corrinne, AK 05708-1071

Phone: +3512198379449

Job: Design Planner

Hobby: Graffiti, Foreign language learning, Gambling, Metalworking, Rowing, Sculling, Sewing

Introduction: My name is Dong Thiel, I am a brainy, happy, tasty, lively, splendid, talented, cooperative person who loves writing and wants to share my knowledge and understanding with you.