The aim in this note is to obtain new hypergeometric forms for the functions (√z − 1 − √z) b ± (√z − 1 − √z)−b, (√z − 1 + √z)b ± (√z − 1 + √z)−b, where b is an arbitrary parameter, in terms of Gauss hypergeometric functions. An application of these results (when b =1/3) is made to obtain the hypergeometric form of the roots of the cubic equation r3 − r + 2/3√2/3 = 0. This complements the entry in the compendium of Prudnikov et al. on page 472, entry (68) of the table, where only the middle root (either real or purely imaginary) is given in hypergeometric form.Publisher's Versio

Bhat, Aarif Hussain

Majid, Javid

Paris, Richard Bruce

Qureshi, Mohammad Idris

TWMS Journal of Applied and Engineering Mathematics

English

Isik University Academic Open Access

TWMS J. Appl. and Eng. Math., V.14, N.1, 2024, pp. 395-401HYPERGEOMETRIC FUNCTION REPRESENTATION OF THEROOTS OF A CERTAIN CUBIC EQUATIONM. I. QURESHI1, R. B. PARIS2, J. MAJID1∗, A. H. BHAT1, §Abstract. The aim in this note is to obtain new hypergeometric forms for the functions(√z − 1−√z)b ± (√z − 1−√z)−b, (√z − 1 +√z)b ± (√z − 1 +√z)−b,where b is an arbitrary parameter, in terms of Gauss hypergeometric functions. An ap-plication of these results (when b = 13) is made to obtain the hypergeometric form ofthe roots of the cubic equation r3 − r + 23√z3= 0. This complements the entry in thecompendium of Prudnikov et al. on page 472, entry (68) of the table, where only themiddle root (either real or purely imaginary) is given in hypergeometric form.Keywords: Gauss hypergeometric function, Roots of a cubic, Pfaff-Kummer transforma-tion.AMS Subject Classification(2020): 33C05, 33C20, 33C99.1. Introduction and preliminariesA remarkable large number of generalized hypergeometric functions and their exten-sions have been given by many authors (see [3, 12]). These functions play a crucial rolein mathematical analysis, engineering and applied sciences. Abdus et al. [9] introduceda new confluent hypergeometric gamma function and an associated confluent hypergeo-metric Pochhammer symbol and studied their many properties like their integral repre-sentations, derivative formulas, and generating function relations. The novel expansion ofbeta function has been given by Musharraf et al.[1] by using multi-index Mittag-Lefflerfunction. Srivastava et al.[11] introduced an extended version of the Pochhammer symboland then introduced the corresponding extension of the τ -Gauss hypergeometric functionand studied their basic properties. Srivastava et al.[10] also gave the extended Pochham-mer symbol by using extended gamma function involving Macdonald function. Further1 Department of Applied Sciences and Humanities, Faculty of Engineering and Technology,Jamia Millia Islamia (A Central University), New Delhi-110025, India.e-mail: miqureshi delhi@yahoo.co.in; ORCID: https://orcid.org/0000-0002-5093-1925.e-mail: javidmajid375@gmail.com; ORCID: https://orcid.org/0000-0001-8916-8384.∗ Corresponding author.e-mail: aarifsaleem19@gmail.com; ORCID: https://orcid.org/0000-0001-9792-1487.2 Division of Computing and Mathematics, Abertay University, Dundee, DD1 1HG, U.K.e-mail: r.paris@abertay.ac.uk; ORCID: https://orcid.org/0000-0001-7868-0278§ Manuscript received: December 30, 2021; accepted: April 03, 2022.TWMS Journal of Applied and Engineering Mathematics, Vol.14, No.1 © Işık University, Departmentof Mathematics, 2024; all rights reserved.395396 TWMS J. APP. AND ENG. MATH. V.14, N.1, 2024the authors gave the corresponding extension of the generalized hypergeometric function.The families of generating functions and generating relations for the extended generalizedhypergeometric function have also been presented.In the book by Prudnikov et al. [4, p. 472, entry(68)] the middle root (either real orpurely imaginary) of the cubic equationr3 − r + 23√z3= 0 (1.1)is given in hypergeometric form by23√z32F1[13 ,2332; z]. (1.2)Here, 2F1(z) denotes the Gauss hypergeometric function defined in terms of the Pochham-mer symbol (a)n = Γ(a+ n)/Γ(a) by2F1[a, bc; z]=∞∑n=0(a)n(b)n(c)nn!zn (|z| < 1)and elsewhere in the complex z-plane cut along [1,∞) by analytic continuation. It is theaim in this note to give hypergeometric representations for the other two roots of (1.1),following recently published work in [2], [5]–[8]. This is achieved by obtaining similarhypergeometric forms for the functions(√z − 1−√z)b ± (√z − 1−√z)−b, (√z − 1 +√z)b ± (√z − 1 +√z)−b,where b is an arbitrary parameter, thereby complementing those expressions given in [4,p. 486].From Mathematica, the roots of the cubic (1.1) can be expressed in the formr1,2(z) = −12√3{(√z − 1−√z)1/3 + (√z − 1−√z)−1/3}± i2{(√z − 1−√z)1/3 − (√z − 1−√z)−1/3}, (1.3)r3(z) =1√3{(√z − 1−√z)1/3 + (√z − 1−√z)−1/3}. (1.4)Elementary considerations show that when 0 ≤ z ≤ 1 the roots are real; when z > 1 thereis one real root and a complex conjugate pair; when z < 0 all three roots are complex withr2(z) being purely imaginary; see Table 1. When z is complex, it is sufficient to consideronly the upper half plane since the roots satisfy rj(z) = rj(z) (1 ≤ j ≤ 3), where the bardenotes the complex conjugate.In what follows we shall make use of the binomial theorem(1− z)−b = 1F0[b− ; z]=∞∑n=0(b)nn!zn, (1.5)where |z| < 1 and b ∈ C, and |z| = 1 when <(b) < 0. We shall also require the Pfaff-Kummer transformation2F1[a, bc; z]= (1− z)−a2F1[a, c− bc;zz − 1](1.6)and Euler’s linear transformation2F1[a, bc; z]= (1− z)c−a−b2F1[c− a, c− bc; z], (1.7)M. I. QURESHI, R. B. PARIS, J. MAJID, A. H. BHAT: HYPERGEOMETRIC FUNCTION... 397Table 1. Values of the roots rj(z) (accurate to 5dp) of the cubic (1.1) for z ∈ [−1, 1].−0.2 −1.01046− 0.08372i 0.16744i 1.01046− 0.08372i−0.4 −1.01983− 0.11555i 0.23109i 1.01983− 0.11555i−0.6 −1.02835− 0.13846i 0.27691i 1.02835− 0.13846i−0.8 −1.03619− 0.15673i 0.31346i 1.03619− 0.15673i−1.0 −1.04347− 0.17207i 0.34414i 1.04347− 0.17207i0 −1.00000 0.00000 1.000000.2 −1.07696 0.17777 0.899210.4 −1.10470 0.26127 0.843430.6 −1.12475 0.33611 0.788640.8 −1.14094 0.41653 0.724401.0 −1.15470 0.57735 0.57735both holding provided c 6= 0,−1,−2, . . . and | arg(1− z)| < π.2. Hypergeometric representation of certain functional formsWe have the following theorem:Theorem 2.1. Let b be an arbitrary parameter and define the two functionsF1(b; z) := 2F1 −12b, 12b ; z12 , F2(b; z) := 2F1 12 − 12b, 12 + 12b ; z32 .Then the following identities hold:(√z − 1−√z)b+(√z − 1−√z)−b = 2{cos(12πb)F1(b; z)− b√z sin(12πb)F2(b; z)}, (2.1)(√z − 1−√z)b − (√z − 1−√z)−b = ±2i{sin(12πb)F1(b; z) + b√z cos(12πb)F2(b; z)},(2.2)(√z − 1+√z)b+(√z − 1+√z)−b = 2{cos(12πb)F1(b; z) + b√z sin(12πb)F2(b; z)}, (2.3)(√z − 1 +√z)b − (√z − 1 +√z)−b = ±2i{sin(12πb)F1(b; z)− b√z cos(12πb)F2(b; z)},(2.4)for z ∈ C\[1,∞). The upper or lower signs in (2.2) and (2.4) are chosen according as0 ≤ arg z ≤ π or arg z < 0, respectively.Proof. For convenience in presentation let us define Z := z/(z−1). Consider the expressionH(b; z) := (√z − 1−√z)b = (z − 1)b/2(1− Z1/2)b = (z − 1)b/21F0 −b; Z−;= (z − 1)b/2{ ∞∑n=0(−b)2n(2n)!Zn +∞∑n=0(−b)2n+1(2n+ 1)!Zn+1/2}398 TWMS J. APP. AND ENG. MATH. V.14, N.1, 2024by (1.5) and in the series decomposition we suppose that |Z| < 1. Employing theidentities(a)2n = 22n(12a)n(12a+12)n, (a)2n+1 = 22na(12a+12)n(12a+ 1)n,we findH(b; z) = (z − 1)b/22F1 −12b, 12 − 12b ;Z12− bZ1/22F1 12 − 12b, 1− 12b ;Z32 .Application of the Pfaff-Kummer transformation (1.6) then yieldsH(b; z) = (z − 1)b/2{(1− z)−b/2F1(b; z)− bZ1/2(1− z)−b/2+1/2F2(b; z)}.Using the fact that√z − 1 = ±i√1− z,{0 ≤ arg z ≤ πarg z < 0,we obtainH(b; z) = e±πib/2(F1(b; z)± ib√zF2(b; z)), (2.5)where the upper or lower sign is taken according as 0 ≤ arg z ≤ π or arg z < 0,respectively. The result (2.5) has been obtained under the assumption that |Z| < 1, butmay be extended by analytic continuation to z ∈ C\[1,∞).Upon noting that Fj(−b; z) = Fj(b; z) (j = 1, 2), it follows immediately upon reversingthe sign of b thatH(−b; z) = e∓iπb/2(F1(b; z)∓ ib√zF2(b; z)). (2.6)Hence some routine algebra producesH(b; z) +H(−b; z) = 2{cos(12πb)F1(b; z)− b√z sin(12πb)F2(b; z) (2.7)andH(b; z)−H(−b; z) = ±2i{sin(12πb)F1(b; z) + b√z cos(12πb)F2(b; z)}(2.8)which are the results stated in (2.1) and (2.3). The proofs of (2.2) and (2.4) follow the same steps and so will be omitted.3. Hypergeometric form of the roots of the cubic (1.1)When b = 13 , we obtain from (2.1)-(2.4) the representations:(√z − 1−√z)1/3 + (√z − 1−√z)−1/3 =√3F1(13 ; z)−√z3 F2(13 ; z),(√z − 1−√z)1/3 − (√z − 1−√z)−1/3 = ±i{F1(13 ; z) +√z3F2(13 ; z)},(√z − 1 +√z)1/3 + (√z − 1 +√z)−1/3 =√3F1(13 ; z) +√z3 F2(13 ; z),(√z − 1 +√z)1/3 − (√z − 1 +√z)−1/3 = ±i{F1(13 ; z)−√z3F2(13 ; z)},(3.1)M. I. QURESHI, R. B. PARIS, J. MAJID, A. H. BHAT: HYPERGEOMETRIC FUNCTION... 399whereF1(13; z) = 2F1 −16 , 16 ; z12 = √1− z2F1 13 , 23 ; z12 , F2(13; z) = 2F1 13 , 23 ; z32 .(3.2)The alternative form of F1(13 ; z) follows from Euler’s transformation (1.7).We then obtain from (2.1) and (2.2) the desired hypergeometric forms of the roots givenin the following theorem:Theorem 3.1.. The roots rj (1 ≤ j ≤ 3) in (2.1) and (2.2) of the cubic equationr3 − r + 23√z3= 0,where 0 ≤ arg z ≤ π, have the hypergeometric representationsr1(z) = −F1(13 ; z)−13√z3F2(13 ; z),r2(z) =23√z3F2(13 ; z),r3(z) = F1(13 ; z)−13√z3F2(13 ; z), (3.3)where the hypergeometric functions F1(13 ; z) and F2(13 ; z) are defined in (3.2). The rootssatisfy rj(z) = rj(z) (1 ≤ j ≤ 3) for conjugate values of z.Proof. The results in (3.3) follow immediately by insertion of the first two formulas in(3.1) (with the upper sign when 0 ≤ arg z ≤ π) into the expressions for the roots rj(z) in(2.1) and (2.2). 4. ConclusionsIn this paper, we have obtained the hypergeometric forms of some new functions (notrecorded earlier) in terms of combinations of Gauss hypergeometric functions by usingseries manipulation and the Pfaff-Kummer linear transformation. In addition, we havealso obtained the hypergeometric form of the two missing roots (not recorded in [4]) ofthe cubic equation (1.1). The various functional forms have all been verified numericallywith the help of Mathematica.We can obtain the hypergeometric forms of the real and complex roots of higher degreepolynomial equations in an analogous manner. Moreover, these results could have potentialapplications in several fields of Applied Mathematics, Statistics and Engineering Sciences.Acknowledgement. The authors are very thankful to the anonymous referees for theirvaluable suggestions to improve the paper in its present form.400 TWMS J. APP. AND ENG. MATH. V.14, N.1, 2024References[1] Ali, M., Ghayasuddin, M., Khan, W. A. and Nisar, K. S., (2021), A novel kind of the multi-index beta,Gauss and confluent hypergeometric functions, Journal of Mathematics and Computer Science, 23, pp.145-154.[2] Bhat, A. H., Qureshi, M. I. and Majid, J., (2020), Hypergeometric forms of certain composite functionsinvolving arcsin(x) using Maclaurin series and their applications, Jñānābha, 50(2), pp. 139-145.[3] Olver F. W. J., Lozier D. W., Boisvert R. F. and Clark C. W. (eds.), (2010), NIST Handbook ofMathematical Functions, Cambridge University Press, Cambridge.[4] Prudnikov A. P., Brychkov Yu. A. and Marichev O. I., (1990), Integrals and Series, Vol. III, Gordonand Breach Science Publishers, New York.[5] Qureshi, M. I., Majid, J. and Bhat, A. H., (2020), Hypergeometric forms of some composite functionscontaining arccos(x) using Maclaurin’s expansion, South East Asian Journal of Mathematics andMathematical Sciences, 16(3), pp. 83-96.[6] Qureshi, M. I., Malik, S. H. and Ara, J., (2020), Hypergeometric forms of some mathematical functionsvia a differential equation approach, Jñānābha, 50(2), pp. 153-159.[7] Qureshi, M. I., Malik, S. H. and Shah, T. R., (2020), Hypergeometric forms of some functions involvingarcsin(x) using a differential equation approach, South East Asian J. of Mathematics and MathematicalSciences, 16 (2), pp. 79-88.[8] Qureshi, M. I., Malik, S. H. and Shah, T. R., (2020), Hypergeometric representations of some mathe-matical functions via Maclaurin series, Jñānābha, 50(1), pp. 179-188.[9] Saboor, A., Rahman, G., Ali, H., Nisar, K. S. and Abdeljawad, T., (2021), Properties and applicationsof a new extended gamma function involving confluent hypergeometric function, Advances in DifferenceEquations, 2021:407, pp. 1-14.[10] Srivastava, H. M., Rahman, G. and Nisar, K. S., (2019), Some Extensions of the Pochham-mer Symbol and the Associated Hypergeometric Functions, Iran J. Sci. Technol. Trans. Sci.,https://doi.org/10.1007/s40995-019-00756-8.[11] Srivastava, H. M., Tassaddiq, A., Rahman, G., Nisar, K. S. and Khan, I., (2019), A New Extensionof the τ -Gauss Hypergeometric Function and Its Associated Properties, Mathematics, 7, 996.[12] Younis, J., Verma, A., Aydi, H., Nisar, K. S. and Alsamir, H., (2021), Recursion formulas for certainquadruple hypergeometric functions, Advances in Difference Equations, 2021:407, pp. 1-14.M. I. QURESHI, R. B. PARIS, J. MAJID, A. H. BHAT: HYPERGEOMETRIC FUNCTION... 401M. I. Qureshi for the photography and short autobiography, see TWMS J. App. and Eng. Math.V.10, N.4.Richard Bruce Paris is an emeritus Reader in Mathematics at the University ofAbertay Dundee, U.K. He also has an honorary appointment at the University of St.Andrews, U.K. Previously, he was employed by Euratom at the French Atomic EnergyCommission at Fontenay-aux-Roses and Cadarache. Paris has published numerouspapers in asymptotics, special functions, and MHD instability theory. Paris waselected a Fellow of the U.K. Institute of Mathematics and its Applications in 1986.Javid Majid received his M.Sc. degree in Mathematics from Jiwaji University,Gwalior, M.P., India in 2014. He received his M.Phil from Jiwaji University, Gwalior,M.P., India in 2018. He is pursuing Ph.D. in Mathematics from the Department ofApplied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Mil-lia Islamia, New Delhi. His research interests are in the areas of Integral Transforms,Special Functions and Multiple Hypergeometric Functions..Aarif Hussain Bhat received his M.Sc. degree in Mathematics from Jiwaji Uni-versity, Gwalior, M.P., India in 2016. He received his M.Phil from Jiwaji University,Gwalior, M.P., India in 2018. He is pursuing Ph.D. in Mathematics from the Depart-ment of Applied Sciences and Humanities, Faculty of Engineering and Technology,Jamia Millia Islamia, New Delhi. His research interests are in the areas of IntegralTransforms, Special Functions and Multiple Hypergeometric Functions.

Hypergeometric function representation of the roots of a certain cubic equation

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Hypergeometric function representation of the roots of a certain cubic equation

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