2017 Volume 25 No.1-2

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1 27 Volume 25 No.-2 GENERAL MATHEMATICS EDITOR-IN-CHIEF Daniel Florin SOFONEA ASSOCIATE EDITOR Ana Maria ACU HONORARY EDITOR Dumitru ACU EDITORIAL BOARD Heinrich Begehr Andrei Duma Dumitru Gaspar Shigeyoshi Owa Dorin Andrica Hari M. Srivastava Malvina Baica Vasile Berinde Piergiulio Corsini Vijay Gupta Gradimir V. Milovanovic Claudiu Kifor Detlef H. Mache Aldo Peretti Adrian Petruşel SCIENTIFIC SECRETARY Emil C. POPA Nicuşor MINCULETE Ioan ŢINCU Augusta RAŢIU EDITORIAL OFFICE DEPARTMENT OF MATHEMATICS AND INFORMATICS GENERAL MATHEMATICS Str.Dr. Ion Ratiu, no Sibiu, ROMANIA Electronical version:

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3 Contents E. L. Pop, D. Duca, A. Raţiu, Properties of the intermediate point from a mean value theorem of the integral calculus S. Deshwal, P.N. Agrawal, Miheşan-Kantorovich operators of blending type A. E. Bărar, Some families of rational Heun functions and combinatorial identities F. Nasaireh, Voronovskaja-type formulas and applications D. Foukrach, On nonexistence of solutions to a nonlinear Cauchy problem for a higher order hyperbolic equation P. Agarwal, C. K. Goel, Extremal sets in a topological space A. Hassanzadeh, A Young s inequality for the Sugeno integrals... 6 A. Kajla, Blending type approximation by summation-integral operators based on Polya distribution E. P. Mazi, T. O. Opoola, On some subclasses of bi-univalent functions associating pseudo-starlike functions with Sakaguchi type functions V. Neagos, A note on the Pompeiu-Stamate mean-value theorem.. 97 M. Iranmanesh, A. G. Sanatee, Lattice g-2-normed spaces and 2-best approximation properties of their downward subsets E. Szatmari, Á. O. Páll-Szabó, Differential subordination results obtained by using a new operator A. Vernescu, About the sequence of general term Ω n = n ) n M. F. Causley, P. Morell, Sequences which converge to e: New insights from an old formula V. Lokesha, S. Jain, T. Deepika, K.M. Devendraiah, Some computational aspects of polycyclic aromatic hydrocarbons

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5 General Mathematics Vol. 25, No ), 3 Properties of the intermediate point from a mean value theorem of the integral calculus Emilia-Loredana Pop, Dorel Duca, Augusta Raţiu Abstract If the functions f, g : [a, b] R satisfy the conditions: a) f and g are continuous on [a, b], b) the function f is decreasing on [a, b], c) f x), for all x [a, b], then there exists a point c [a, b] such that b a c f x) g x) dx = f a) a g x) dx. In this paper, we study the approaching of the point c towards a, when b approaches a. 2 Mathematics Subject Classification: 26A24. Key words and phrases: intermediate point, mean-value theorem. Introduction and Preliminaries In the proof of the second mean-value theorem of integral calculus Bonnet s theorem) one uses the following theorem. Theorem Let a, b R with a < b and f, g : [a, b] R. If a) the function f is decreasing on [a, b], b) f x), for all x [a, b], c) the function g is Riemann integrable on [a, b], then the function fg is Riemann integrable on [a, b] and there exists a point c [a, b] such that b c f x) g x) dx = f a) g x) dx. a a Received September, 27 Accepted for publication in revised form) 2 October, 27 3

6 4 E. L. Pop, D. Duca, A. Raţiu For the proof of this theorem, one can see [2]. It follows that if the continuous functions f, g : [a, b] R satisfy the following properties: a) the function f is decreasing on [a, b], b) f x), for all x [a, b], then, for each x a, b], there exists a point c x [a, x] such that ) x a f t) g t) dt = f a) cx a g t) dt. If for each x a, b] there exists a unique point c x [a, x] such that ) holds, we can define the function c : a, b] [a, b] by 2) cx) = c x, for all x a, b]. The function c has the property that 3) x a cx) f t) g t) dt = f a) a g t) dt, for all x a, b]. If for some x a, b], there exist more points c x [a, x] such that 3) is true, then for each x a, b] choose c x [a, x] which satisfies 3). It follows that we can also define the function c : a, b] [a, b] by formula 2). This function c satisfies 3), too. Consequently, the following statement is true. Theorem 2 Let a, b R with a < b.if the continuous functions f, g : [a, b] R satisfy properties: a) the function f is decreasing on [a, b], b) f x), for all x [a, b], then there exists a function c : a, b] [a, b] such that 3) is true. If x a, b] tends to a, because c x) a x a, we have limc x) = a. x a Then the function c : [a, b] [a, b] defined by { c x), if x a, b] c x) = a, if x = a. is continuous at x = a. The purpose of this paper is to establish under which circumstances the function c is the third order differentiable at the point x = a and to compute its derivatives c ) a), c 2) a) and c 3) a). Do the derivatives c a), c a) and c a) depend upon the functions f and g? If there exist several functions c which satisfy 3), do the derivatives of the function c at x = a depend upon the function c we choose?

7 The intermediate point from a mean value theorem 5 Since for x a, b], if we denote by then and and hence c x) c a) x a = c x) a x a, θ x) = c x) a x a, θ x) [, ] c x) = a + x a) θ x) 4) x a a+x a)θx) f t) g t) dt = f a) a Consequently, the following statement is true. g t) dt, for all x a, b]. Theorem 3 Let a, b R with a < b.if the continuous functions f, g : [a, b] R satisfies proprties: a) the function f is decreasing on [a, b], b) f x), for all x [a, b], then there exists a function θ : a, b] [, ] such that 4) is true. Another purpose of this paper is to establish in which conditions the function θ : a, b] [, ] has limit in the point x = a and also in which conditions the extension through continuity of the function θ, denoted by θ, is derivable of first and second order in the point x = a. The study of these functions, for other mean value theorems, can be find in papers like [3], [4], [5], [6], [7]. 2 Main results First, we remark that F is a primitive of the function fg and G is a primitive of the function g, then, from the Leibniz Newton Theorem, the equality 4) becomes 5) F x) F a) = fa)[ga + x a)θx)) Ga)], for all x a, b]. In this section we present some results related to the intermediate point functions c and θ. Theorem 4 Let a, b R with a < b and let f, g be two continuous functions on [a, b]. If a) the function f is decreasing on [a, b], b) f x), for all x [a, b],

8 6 E. L. Pop, D. Duca, A. Raţiu d) fa)ga), then The function θ : a, b] [, ] has limit at the point x = a and lim x a θx) =. 2 The function c is differentiable at x = a and c a) =. Proof. Equality 5) can be write as follows 6) F x) F a) x a = fa) Ga+x a)θx)) Ga) θx), x a)θx) for all x a, b]. Obviously, F x) F a) lim = F a) = fa)ga). x a x a Because the function θ is bounded, we have that and hence lim x a)θx) =, x a lim x a Ga+x a)θx)) Ga) x a)θx) = lim x a Ga+x a)θx)) Ga) a+x a)θx) a = G a) = ga). It follows that, if f a) ga) then there exists limθx) and from 6) we obtain x a that fa)ga) = fa)ga) lim θx), x a so lim x a θx) =. 2 The statement 2 follows from the statement. In what follows, we denote by θ : [a, b] [, ] de function defined by { θ x), if x a, b] θ x) =, if x = a. Theorem 5 Let a, b R with a < b and let f, g be differentiable functions on [a, b]. If a) the function f is decreasing on [a, b], b) f x), for all x [a, b], c) f and g are continuous at x = a,

9 The intermediate point from a mean value theorem 7 d) fa)ga), then The function θ is differentiable at x = a and θ a) = f a) 2fa). 2 The function c is twice differentiable at x = a and c a) = f a) 2fa). Proof. By Taylor s theorem, for each x a, b], there exist ξ x, η x a, x) such that F x) = F a) + F a)! Gx) = Ga) + G a)! We replace these formulas in 5) and get F a)! + F ξ x ) 2! x a) = fa) [ G a)! x a) + F ξ x ) x a) 2, 2! x a) + G η x ) x a) 2. 2! θx) + G η x ) x a)θ 2 x) ]. 2! For each x [a, b], we have F x) = f x) g x), F x) = f x) g x) + f x) g x), G x) = g x), G x) = g x), so the last equality becomes fa)ga)! + f ξ x)gξ x)+fξ x)g ξ x) 2! x a) = fa)ga)! θx)+ which is equivalent with [ ] fa)ga) θx) x a = 2! + fa)g η x) 2! x a)θ 2 x), [ ] f ξ x )gξ x ) + fξ x )g ξ x ) fa)g η x )θ 2 x), for all x a, b] Since lim θx) = and f, f, g, g are continuous at x = a, it follows that x a [ ] lim f ξ x )gξ x ) + fξ x )g ξ x ) fa)g η x )θ 2 x) = x a 2! 2 f a) g a), and then there exists and Hence θ θx) θ a) θx) a) = lim = lim x a x a x a x a fa)ga)θ a) = 2 f a)ga). θ a) = f a) 2fa). 2 The statement 2 follows from the statement.

10 8 E. L. Pop, D. Duca, A. Raţiu Theorem 6 Let a, b R with a < b and let f, g be two twice differentiable functions on [a, b]. If a) the function f is decreasing on [a, b], b) f x), for all x [a, b], c) f and g are continuous at x = a, d) fa)ga), then The function θ is twice differentiable at x = a and θ a) = f a)ga) f a)g a). 3fa)ga) 2 The function c is third differentiable at x = a and c a) = f a)ga) f a)g a). 3fa)ga) Proof. By Taylor s theorem, for each x a, b], there are ξ x, η x a, x) such that F x) = F a) + F a)! x a) + F a) 2! x a) 2 + F ξ x ) x a) 3 3! Gx) = Ga) + G a) x a) + G a) x a) 2 + G η x ) x a) 3.! 2! 3! We replace these formulas in 5) and get F a)! + F a) 2! x a) + F ξ x) 3! x a) 2 = = fa) [ G a)! θx) + G a) 2! x a)θ 2 x) + G η x) 3! x a) 2 θ 3 x) ]. For each x [a, b], we have F x) = f x) g x), F x) = f x) g x) + f x) g x), F x) = f x)gx) + 2f x)g x) + fx)g x), G x) = g x), G x) = g x), G x) = g x), so the last equality becomes fa)ga) θx) + 3! x a = 2! [ f a)ga) + fa)g a) fa)g a)θ 2 x) ] + [ f ξ x )gξ x ) + 2f ξ x )g ξ x ) + fξ x )g ξ x ) fa)g η x )θ 3 x) ] x a), or, equivalently fa)ga) + 2! + 3! [ θx) θa) x a ] θ a) = f a)ga) 2 + [ f a)ga) + fa)g a) fa)g a)θ 2 x) [ ] f ξ x )gξ x ) + 2f ξ x )g ξ x ) + fξ x )g ξ x ) fa)g η x )θ 3 x) x a), ] +

11 The intermediate point from a mean value theorem 9 and then the last equality becomes fa)ga) + 3! θx) θa) θ x) x a x a = θx) θa) x a fa)g a)[ ] 2 θx) + + [ f ξ x )gξ x ) + 2f ξ x )g ξ x ) + fξ x )g ξ x ) fa)g η x )θ 3 x) ]. Since lim θ x) = θ a) =, lim x a x a θ x) θ x) x a = θ a) = f a) 2f a), it follows that the function θ is twice differentiable at x = a, and fa)ga) θ a) 2 = fa)g a) f a) 2 2fa) + )+ [ ] f a)ga) + 2f a)g a) + fa)g a) fa)g a). + 3! Hence, or fa)ga)θ a) = f a)ga) f a)g a), 3 θ a) = f a)ga) f a)g a). 3fa)ga) 2 The statement 2 follows from the statement. 3 Conclusions and further challenges In this paper we gave conditions for the functions f and g such that the intermediate point functions c and θ to be derivable in the point a and also we provided the derivative c a), θ a), c a), θ a) and c a). In future we want to verify in which conditions the functions c and θ are differentiable of order n in the point a and to calculate the corresponding derivative. References [] D. I. Duca, Analiză Matematică, Casa Cărţii de Ştiinţă, Cluj-Napoca, vol., 23. [2] D. I. Duca, Analiză Matematică, vol. 2, to appear. [3] D. I. Duca, A Note on the Mean Value Theorem, Didactica Matematica, vol. 9, 23, 9-2. [4] D. I. Duca, Properties of the intermediate point from the Taylor s theorem, Mathematical Inequalities & Applications, vol. 2, no. 4, 29,

12 E. L. Pop, D. Duca, A. Raţiu [5] D. I. Duca, O. Pop, On the Intermediate Point in Cauchy s Theorem, Mathematical Inequalities & Applications, vol. 9, no. 3, 26, [6] D. I. Duca, O. Pop, Concerning the Intermediate Point in the Mean-Value Theorem, Mathematical Inequalities & Applications, vol. 2, no. 3, 29, [7] T. Trif, Asymptotic Behavior of Intermediate Points in certain Mean Value Theorems, Journal of Mathematical Inequalities, vol. 2, no. 2, 28, 5-6. Emilia-Loredana Pop Babeş-Bolyai University Faculty of Mathematics and Computer Science Department of Computer Science Street Mihail Kogălniceanu, 484 Cluj-Napoca, Romania pop emilia loredana@yahoo.com Dorel Duca Babeş-Bolyai University Faculty of Mathematics and Computer Science Department of Mathematics Street Mihail Kogălniceanu, 484 Cluj-Napoca, Romania dorelduca@yahoo.com Augusta Raţiu Lucian Blaga University of Sibiu Faculty of Science Department of Mathematics and Computer Science Street Dr. I. Raţiu 5-7, 552 Sibiu, Romania augu23@yahoo.com

13 General Mathematics Vol. 25, No ), 27 Miheşan-Kantorovich operators of blending type Sheetal Deshwal, P. N. Agrawal Abstract Miheşan 28) generalized the Szász operators by introducing a general class of linear positive operators. We introduce a Kantorovich generalization of Miheşan operator based on a real parameter ρ. We study the approximation properties of these operators including weighted Korovkin theorem, the rate of convergence in terms of the modulus of continuity, second order modulus of continuity via Steklov-mean, the degree of approximation for the weighted spaces. Furthermore, we obtain the rate of convergence of the considered operators with the aid of the Ditzian-Totik modulus of smoothness and K-functional. At the last we investigate quantitative Voronovskaja and Grüss-Voronovskaja type theorems. 2 Mathematics Subject Classification: 4A, 4A25, 4A28, 4A35, 4A36. Key words and phrases: Linear positive operators, Miheşan operators, unified Ditzian-Totik modulus of smoothness, weighted spaces. Introduction In 28, Miheşan [2],Theorem 4.2) obtained a vital generalization of the Szász operators by using Gamma transformation depending on real parameter η as.) M η αf)x) = ) m η k α,k x)f, x [, ), α k= αx η )k with η + αx >, m η α,k x) = η) k k! well known Pochhammer symbol defined as Received 5 June, 27 Accepted for publication in revised form) 3 August, 27 + αx η )η+k where η may depend on α and a) k is

14 2 S. Deshwal, P.N. Agrawal a) k = aa + )... a + k ), a) =. Many researchers studied Szász-Mirakyan operators and their modifications. Varma and Taşdelen [25], introduced a generalization of Szász operators involving Charlier polynomials. In 26, Ozarslan and Duman [2], proposed a modified Bernstein- Kantorovich operator based on a parameter ρ >,.2) K α,ρ f; x) = α p α,k x) k= ) k + t ρ f dt, x [, ], α + where p α,k x) = α k) x k x) α k. In view of operators.) and.2), for each h C γ [, ) := {f C[, ) : ft) M f + t γ ), M f is a constant depending only on function f)} we construct the following sequence of linear positive operators of blending type.3) M η α,ρh; x) = ) m η k + u ρ α,k x) f du, α + k= The operator defined by.3) is named as Miheşan-Kantorovich operator of blending type. The approximation properties of different mixed hybrid operators has been studied by many researchers in last few years. In [22], Srivastava and Gupta investigated approximation properties of certain family of summation integral-type operators. After that, Yüksel and Ispir [26] studied weighted approximation properties of certain family of summation integral-type operators. Agrawal et al. [5], introduced the Baskakov-Szász type operator depending on a nonnegative parameter and studied rate of convergence and simultaneous approximation properties. In [5], Gupta and Russias studied approximation properties of Durrmeyer type generalization of Szász type operators. In [4], Gupta defined a sequence of blending type operators involving the weights of Pǎltǎnea basis function and investigated the rate of convergence of these operators. In 26, Tuncer and Ulusoy [3], studied approximation properties of mixed Szász-Durrmeyer type operators. Very recently Kajla et al. [7], introduced the hybrid operators based on inverse Polya-Eggenberger distribution and studied their rate of convergence. Kajla and Araci [8], studied approximation properties of blending type Stancu-Kantorovich operators based on Polya-Eggenberger distribution. In this present work, our focus is to study the approximation properties of Miheşan- Kantorovich operator.3) via Bohman-Korovkin theorem and in terms of first and second order modulus of continuity and Steklov mean. We estimate the rate of convergence of these operators in terms of Ditzian-Totik modulus of smoothness and K-functional. Furthermore, we investigate weighted approximation theorems. Lastly we study quantitative Voronovskaja and Grüss-Voronovskaja type theorem.

15 Miheşan-Kantorovich operators of blending type 3 2 Preliminaries Throughout this paper, we assume that η = ηα), as α and α lim α = l R). In the following lemma we obtain the values of the moments for the operators M η ηα) α,ρ. Lemma The operators M η α,ρ satisfies following equalities: i) M η α,ρ; x) =, ii) M η α,ρu; x) = αx α + + ρ + )α + ), iii) M η α,ρu 2 ; x) = α2 x 2 η + ) αxρ + 3) ηα + ) 2 + ρ + )α + ) 2 + 2ρ + )α + ) 2, iv) M η α,ρu 3 ; x) = α3 x 3 + η)2 + η) ηα + ) 3 + 3x2 α 2 + η)ρ + 2) ηρ + )α + ) 3 αxρ + )2ρ + ) + 33ρ + 2)) + α + ) 3 + ρ + )2ρ + ) 3ρ + )α + ) 3, v) M η α,ρu 4 ; x) = α4 x 4 + η)2 + η)3 + η) η 3 α + ) 4 + α3 x 3 + η)2 + η)6ρ + ) η 2 ρ + )α + ) 4 + α2 x 2 [7 + η)ρ + )2ρ + ) +6ηρ + ) + 6ρ + )+ 2 + η)2ρ + )] ηρ + )2ρ + )α + ) 4 + α + ) 4 [αx{ρ + )2ρ + )3ρ + ) + 6ρ + )3ρ + ) ρ + )2ρ + )3ρ + ) + 42ρ + )3ρ + ) + 4ρ + )2ρ + )}] + 4ρ + )α + ) 4, vi) M η α,ρu 5 ; x) = α5 x 5 + η)2 + η)3 + η)4 + η) η 4 α + ) 4 + α4 x 4 + η)2 + η)3 + η) η 3 α + ) 5 ρ + ) α 3 x 3 + η)2 + η) ρ + 5) + η 2 α + ) 5 [25ρ + )2ρ + ) + 32ρ + ) ρ + )2ρ + ) α 2 x 2 + η) + ρ + )] + ηα + ) 5 [5ρ + )2ρ + )3ρ + ) ρ + )2ρ + )3ρ + ) + 352ρ + )3ρ + ) + 3ρ + )3ρ + ) + ρ + )2ρ + )] αx + α + ) 5 [ρ + )2ρ + )3ρ + )4ρ + ) ρ + )2ρ + )3ρ + )4ρ + ) + 52ρ + )3ρ + )4ρ + ) + ρ + )3ρ + )4ρ + ) + ρ + )2ρ + ) 4ρ + ) + 5ρ + )2ρ + )3ρ + )] + 5ρ + )α + ) 5,

16 4 S. Deshwal, P.N. Agrawal vii) M η α,ρu 6 ; x) = α6 x 6 + η)2 + η)3 + η)4 + η)5 + η) η 5 α + ) 6 + α5 x 5 + η)2 + η)3 + η)4 + η)5ρ + 2) η 4 α + ) 6 ρ + ) + α4 x 4 + η)2 + η)3 + η) η 3 α + ) 6 ρ + )2ρ + ) [65ρ + )2ρ + ) + 62ρ + ) + 5ρ + )] α 3 x 3 + η)2 + η) + α + ) 6 η 2 ρ + )2ρ + )3ρ + ) [9ρ + )2ρ + )3ρ + ) + 52ρ + )3ρ + ) + 9ρ + )3ρ + ) + 2ρ + ) α 2 x 2 + η) 2ρ + )] + α + ) 6 [3ρ + )2ρ + ) ηρ + )2ρ + )3ρ + )4ρ + ) 3ρ + )4ρ + ) + 92ρ + )3ρ + )4ρ + ) + 5ρ + )3ρ + )4ρ + ) + 6ρ + ) 2ρ + )4ρ + ) + 5ρ + )2ρ + )3ρ + )] αx + α + ) 6 [ρ + )2ρ + ) ρ + )2ρ + )3ρ + )4ρ + )5ρ + ) 3ρ + )4ρ + )5ρ + ) + 62ρ + )3ρ + )4ρ + )5ρ + ) + 5ρ + )3ρ + ) 4ρ + )5ρ + ) + 2ρ + )2ρ + )4ρ + )5ρ + ) + 5ρ + )2ρ + ) 3ρ + )5ρ + ) + 6ρ + )2ρ + )3ρ + )4ρ + )] + 6ρ + )α + ) 6. Proof. Using Lemma 2. from [6] and simple calculations, the identities i)-vii) are proved. Hence we skip the details. As a consequence of Lemma, we obtain: Lemma 2 The operator M η α,ρ verifies the following equalities: i) M η α,ρu x; x) = x α + + α + )ρ + ), ii) M η α,ρu x) 2 ; x) = x2 α 2 + η) xαρ + α 2) + ηα + ) 2 α + ) 2 ρ + ) + 2ρ + )α + ) 2, iii) M η α,ρu x) 4 ; x) = x 3 x 4 η 3 α + ) 4 [α4 3η + 6) 8ηα 3 + 6α 2 η 2 + η 3 ] + η 2 α + ) 4 ρ + ) [α3 6ηρ + 6η + 2ρ + 2) 2ηα 2 ρ + ) + 6η 2 ρ + ) x 2 4η 2 ] + α + ) 4 ηρ + )2ρ + ) [α2 {ρ 2 6η + 4) + ρ9η + 5) + 3η + 25)} x α{8ηρ + )ρ + 2)} + 6ηρ + )] + α + ) 4 ρ + )2ρ + )3ρ + ) [α{6ρ ρ 2 + 5ρ + 42ρ 2 + 3ρ + )}] + α + ) 4 4ρ + ) ; iv) M η α,ρu x) 6 ; x) = x 6 α + ) 6 η 5 [α6 5η 2 + 3η + 2) α 5 2η η) + α 4 45η 3 + 9η 2 ) 4α 2 η 3 + 5α 2 η 4 + η 5 ] + x 5 α + ) 6 η 4 ρ + )

17 Miheşan-Kantorovich operators of blending type 5 [α 5 {ρ45η η + 36) + 45η 2 + 5η + 54)} α 4 {ρ3η η) + 39η η)} + α 3 {ρ9η 3 + 8η 2 ) + 9η 3 + 3η 2 )} α 2 6η 3 ρ + 2η 3 ) + α{5ρη 4 + 5η 4 } 6η 4 ] x 4 + α + ) 6 η 3 ρ + )2ρ + ) [α4 {ρ 2 9η η + 78) + ρ35η η + 98) + 45η η + 84)} α 3 {ρ 2 48η 2 + 6η) + ρ8η η) + 42η η)} + α 2 {ρ9η 3 + 2η 2 ) + ρ35η η 2 ) + 45η η 2 )} α{4η 3 ρ 2 + 8η 3 ρ + 8η 3 } + 5η 3 x 3 ] + α + ) 6 η 2 ρ + )2ρ + )3ρ + ) [α 3 {ρ 3 9η 2 + 8η + 8) + ρ 2 65η η + 44) + ρ9η η + 342) + 5η 2 + 5η + 7)} α 2 {ρ 3 36η η) + ρ 2 2η η) + ρ8η η) + 5η η)} + α{9η 2 ρ η 2 ρ η 2 ρ + 65} 2η 2 x 2 ] + α + ) 6 ηρ + )2ρ + )3ρ + )4ρ + ) [α 2 {ρ 2 6η +744)+ρ 3 323η +554)+ρ 2 329η +6425)+ρ5η +247)+ 3η + 3)} α{44ηρ ρ 3 η + 23ηρ + 56} + 5η] x + α + ) 6 ρ + )2ρ + )3ρ + )4ρ + )5ρ + ) [α{2ρ ρ ρ ρ ρ+57} 44ρ 4 +3ρ 3 +25ρ 2 + 6ρ + 6)] + α + ) 6 6ρ + ). Lemma 3 For every x [, ), we have i) lim α α M η α,ρu x); x) = x + ρ + ; ii) lim α α M η α,ρu x) 2 ; x) = xlx + ); iii) lim α α 2 M η α,ρu x) 4 ; x) = 3x 2 lx + ) 2 ; iv) lim α α 3 M η α,ρu x) 6 ; x) = 5x 3 lx + ) 3. Consequently, for every x [, ) and sufficiently large α, we can find a constant c = cl) > such that M η α,ρu x) 2 ; x) c + x2 ), M η α,ρu x) 4 ; x) c + x2 ) 2 α α 2 M η α,ρu x) 6 ; x) c + x2 ) 3 α 3. Proof. The proof of Lemma 3 follows easily from Lemma 2, so the details are omitted. 3 Main results In the following theorem we show that the operators M η α,ρ is an approximation process for functions in C γ R + ). and

18 6 S. Deshwal, P.N. Agrawal Theorem Let h C γ R + ). Then, lim α Mη α,ρh; x) hx), uniformly on each compact subset A of [, ). Proof. In view of Lemma, M η α,ρu i ; x) x i, as α, uniformly on A, for i =,, 2. Hence, the required result follows on applying the Bohman-Korovkin criterion. Let C B [, ) denote the space of bounded and uniformly continuous functions on [, ) endowed with the sup norm, f = sup x [, ) fx). The first and second order modulus of continuity are respectively defined as and ω 2 f, δ) = ωf, δ) = sup fx + u) fx + v) x,u,v [, ), u v δ sup fx + 2u) 2fx + u + v) + fx + 2v), δ >. x,u,v [, ), u v δ In our next result we obtain the degree of approximation in terms of the modulus of continuity. Theorem 2 Let h C B [, )and ωh; δ), δ >, be its first order modulus of continuity. Then the operator M η α,ρ satisfies the inequality + M η α,ρh; x) hx) x 2 α 2 ) + η) xαρ + α 2) + ηα + ) 2 α + ) 2 ρ + ) + 2ρ + )α + ) 2 ω h, ). α Proof. By definition of ωh; δ), Lemma 2 and Cauchy-Schwarz inequality, we may get M η α,ρh; x) hx) M η α,ρ hu) hx) ; x) + ) δ Mη α,ρ u x ; x) ωh; δ) + ) M η α,ρu x) δ 2 ; x) ωh; δ) + δ ωh; δ). Now, choosing δ = α /2, we arrive to result. For f C B [, ), the Steklov mean is defined as 3.) f h x) = 4 h 2 h 2 h 2 x 2 α 2 +η) ηα + ) 2 + xαρ+α 2) ) α+) 2 ρ+) + 2ρ+)α+) 2 [2fx + u + v) fx + 2u + v))]dudv.

19 Miheşan-Kantorovich operators of blending type 7 Lemma 4 [3] The Steklov mean f h x) satisfies the following properties: i) f h f ω 2 f, h), ii) f h, f h C B[, ) and f h 5 ωf, h), f h h 9 h 2 ω 2f, h). By using Steklov mean, in our next theorem we establish the rate of convergence in terms of the first and second order modulus of continuity. Theorem 3 Let h C B [, ). Then for each x [, ), we have M η α,ρh; x) hx) 5 x + α ρ + ) { x 2 α 2 + η) + ηα + ) + 2ρ + )α + ) ) ωh; α + ) /2 ) xαρ + α 2) α + )ρ + ) }) ω 2 h; α + ) /2 ). Proof. Applying Lemma and Lemma 4, one has M η α,ρh f h ; x) h f h ω 2 h; h). Since f h C B[, ), by Taylor s expansion, u f h u) = f h x) + u x)f h x) + u s)f h s)ds. x Applying operator M η α,ρ on the above equality, we get M η α,ρf h u) f h x); x) f h Mη α,ρu x; x) + f h 2 Mη α,ρu x) 2 ; x).

20 8 S. Deshwal, P.N. Agrawal Hence using Lemma 2 and Lemma 4, we have M η α,ρh; x) hx) M η α,ρh f h ; x) + M η α,ρf h f h x); x) + f h x) hx) ) ω 2 h; h) + f x h α α + )ρ + ) x 2 α 2 + η) xαρ + α 2) + ηα + ) 2 α + ) 2 ρ + ) ) + 2ρ + )α + ) 2 + f h h 5 ) x h α + + ωh; h) + α + )ρ + ) { x 2 α 2 + η) xαρ + α 2) + ηα + ) 2 α + ) 2 ρ + ) }) + 2ρ + )α + ) 2 ω 2 h; h) h 2 f h 2 Finally, choosing h = α + ) /2, the required result is obtained. In the following theorem, we obtain the estimate of error in approximation for continuously differentiable functions. Theorem 4 For h C B [, ), we have M η α,ρh; x) hx) M x α + + α + )ρ + ) + ωh ; M η α,ρu x)) 2 ; x) + M η α,ρu x) 2 ; x)), where M is some positive constant, and ωh ; δ) denotes the modulus of continuity of h. Proof. Since h C B [, ) M > such that h x) M, x. Using mean value theorem, one may write hu) = hx) + u x)h ζ) = hx) + u x)h x) + u x)h ζ) h x)), where ζ lies between u and x. Applying the operator M η α,ρ on both sides of the above equality and using Lemma 2, we get M η α,ρh; x) hx) h x) M η α,ρu x; x) + M η α,ρ u x h ζ) h x) ; x) M x α + + α + )ρ + ) + Mη α,ρ u x h ζ) h x) ; x). 3.2)

21 Miheşan-Kantorovich operators of blending type 9 3.3) Now, applying Cauchy-Schwarz inequality, we can get u x ωh, δ) + M η α,ρ u x h ζ) h x) ; x) M η α,ρ ωh, δ)m η α,ρ u x + ωh, δ) M η α,ρu x) 2 ; x) + ωh, δ) M η δ α,ρu x) 2 ; x). ) ) u x ; x δ ) u x)2 ; x δ Choosing δ = M η α,ρu x) 2 ; x) and combining 3.2)-3.3), we arrive to conclusion. Now, we recall the definition of second order Ditzian-Totik modulus see[8])as: ωφ 2 h, δ) = sup sup t δ x+tφx) [, ) hx + tφx))) 2hx) + hx tφx)), δ, where φ : [, ) R is an admissible step-weight function. The corresponding K-functional is where K 2,φ h, δ) = inf{ h g + δ φ 2 g, g W 2 φ)}, δ, W 2 φ) = {g C B [, ) : g AC loc [, ).φ 2 g C B [, )}. Also, the Ditzian-Totik modulus of the first order is given by ω φ h, δ) = sup sup t δ x±tφx) [, ) hx + tφx) hx)). From [8],for an absolute constant C, the following relation between second order Ditzian-Totik modulus and K-functional is well known C ω 2 φ h, δ) K 2,φ h, δ) Cω 2 φ h, δ). In order to prove our next result, we consider φ 2 x) = + x 2. In the following theorem, we establish the rate of convergence with the aid of the Ditzian-Totik modulus of the first and second order. α Theorem 5 Let h C B [, ), ηα) as α, and ηα) l R as α. Then for sufficiently large α, there is an absolute constant c such that M η α,ρh; x) hx) 4K 2,φ h, ) c + ω φ h, 2α ) ) c c Cωφ 2 h, + ω φ h, α 2α ) c. α

22 2 S. Deshwal, P.N. Agrawal Proof. First we define auxiliary operator as αx 3.4) M η α,ρh; x) = M η α,ρh; x) h α + + ρ + )α + ) Using definition of the operator M η α,ρ, it is easy to verify that M η α,ρu x; x) =. Now, for any g W φ), 2 using Taylor s formula, we have gu) = gx) + g x)u x) + u x g s)s x)ds. ) + hx). M η α,ρ; x) = and Applying operator M η α,ρ on above equality, we obtain M η α,ρg; x) gx) g x) M η u α,ρu x); x) + M η α,ρ g s)s x)ds; x) x u ) M η α,ρ g s)s x)ds ; x From Lemma 3, + x x αxρ + ) + ρ + )α + ) ) αx α + + ρ + )α + ) s φ2 g φ 2 x) Mη α,ρu x) 2 ; x) + φ2 g φ 2 x) ) αx α ρ + )α + ) x 2 φ2 g {M ηα,ρu φ 2 x) 2 ; x) + M ηα,ρu } x); x)) 2. x) g s)ds therefore, M η α,ρu x) 2 ; x) φ 2 x) c α and Mη α,ρu x); x)) 2 φ 2 x) c α 2, Now, we have M η α,ρg; x) gx) 2c α φ2 g. M η α,ρh; x) hx) M η α,ρh g; x) h g)x) + M η α,ρg; x) gx) ) + αx h α + + hx) ρ + )α + ) 4 h g + 2c α φ2 g ) + αx h α + + hx) ρ + )α + ) 3.5).

23 Miheşan-Kantorovich operators of blending type 2 Also, we have ) αx h α + + hx) ρ + )α + ) ) = x α h α+ )x + ) ρ+)α+) + φx) hx) φx) ) sup h x + φx) Mη α,ρu x; x) hx) φx) ) c 3.6) ω φ h, α Therefore, from 3.5) and 3.6), it follows that { M η α,ρh; x) hx) 4 h g + c } 2α φ2 g + ω φ h, ) c. α Now using the definition of K 2,φ, we immediately arrive to the required result. Next, we define some weighted spaces on [, ) to investigate the weighted approximation results for the operators defined by.3). B σ R + ) := {f : fx) M f σx)}, C σ R + ) := {f : f B σ R + ) C[, ))}, and C k σr + ) := { f : f C σ R + ) and } fx) lim = k some constant)), x σx) where σx) = + x 2 is a weight function and M f is a constant depending only on the function f. From [6], it is noted that C σ R + ) is a normed linear space endowed fx) with the norm f σ := sup x σx). It is well known that the classical modulus of continuity ωf; δ) does not tend to zero if f is continuous on an infinite interval. Therefore, in order to study the approximation of functions in the weighted space CσR k + ), Ispir and Atakut [6] introduced the following weighted modulus of continuity 3.7) Ωf; δ) = fx + h) fx) sup x [, ), h δ + h 2 ) + x 2 ), and proved that lim δ Ωf; δ) =, ωf; λδ) 2 + λ) + δ 2 )Ωf; δ), λ >, and

24 22 S. Deshwal, P.N. Agrawal 3.8) fu) fx) 2 + u x δ u, x [, ). ) + δ 2 ) + x 2 ) + u x) 2 )Ωf; δ), In the following theorem we show that the operator M η α,ρ is an approximation method for functions belonging to the weighted space C k σr + ): Theorem 6 For each h C k σr + ), the sequence of linear positive operators {M η n,ρ}, satisfies following equality lim α Mη α,ρh; x) hx) σ =. Proof. From Lemma, clearly lim α M η α,ρ; x) σ =. Now, M η α,ρu; x) x sup x + x 2 α + sup x x + x 2 + ρ + )α + ) sup x 2α + ) + ρ + )α + ). + x 2 Therefore, lim α M η α,ρu; x) x σ =. Again, M η α,ρu 2 ; x) x 2 sup x + x 2 α2 2αη η ηα + ) 2 sup x sup x x 2 + x 2 + αρ + 3) ρ + )α + ) 2 x + x 2 + 2ρ + )α + ) 2 sup x α2 2αη η αρ + 3) ηα + ) 2 + 2ρ + )α + ) 2 + 2ρ + )α + ) 2, + x 2 we obtain lim α M η α,ρu 2 ; x) x 2 σ =. Hence, applying weighted Korovkintype theorem given by Gadzhiev [9], we reach the desired result. In the following theorem, the rate of convergence is obtained by means of the weighted modulus of continuity. Theorem 7 Let h CσR k + ). Then for sufficiently large α, the following inequality is verified M η α,ρh; x) hx) sup x [, ) + x 2 ) 5/2 KΩ h; ), α where K is a constant not dependent on h and α.

25 Miheşan-Kantorovich operators of blending type 23 Proof. Using the definition of Ωf; δ), Lemma 3 and Cauchy-Schwarz inequality, one can easily see that M η α,ρh; x) hx) M η α,ρ hu) hx) ; x) 2 + δ 2 ) + x 2 )Ωh; δ) ) ) M η u x α,ρ + + u x) 2 ); x δ 2 + δ 2 ) + x 2 )Ωh; δ) M η α,ρ; x) +M η α,ρu x) 2 ; x) + δ Mη α,ρu x) 2 ; x)) /2 + δ Mη α,ρu x) 2 ; x)) /2 M η α,ρu x) 4 ; x)) /2 ) Now, choosing δ = α, we arrive to conclusion immediately. In the following result, we prove a quantitative Voronovskaja type theorem by utilizing the weighted modulus of continuity. Theorem 8 Let h CσR k + ) such that h, h CσR k + ). Then, for any x [, ), we have following equality α h x) Mη h x) α,ρh; x) hx) xρ + )) α + )ρ + ) 2α + ) 2 x 2 α 2 + η) xαρ + α 2) + + η ρ + ) 2ρ + )) = 8α + x 2 )Ωh ; δ)o). Proof. For every h, h C k σ[, ) and u < ζ < x, by Taylor s expansion, we have 3.9) where Λu, x) is given by hu) = hx) + h x)u x) + h ζ) u x) 2 2! = hx) + h x)u x) + h x) u x) 2 + Λu, x), 2! Λu, x) = h ζ) h x) u x) 2. 2! Applying operator M η α,ρ on equation 3.9), we obtain Mη α,ρh; x) hx) h x)m η α,ρu x; x) h 2! Mη α,ρu x) 2 ; x) 3.) M η α,ρλu, x); x).

26 24 S. Deshwal, P.N. Agrawal By the definition 3.7) of weighted modulus of continuity, Λu, x) 2! Ωh ; ζ x ) + ζ x) 2 ) + x 2 )u x) 2 3.) 2! Ωh ; ζ x ) + ζ x) 2 ) + x 2 )u x) 2 ) u x + + δ 2 )Ωh ; δ) + u x) 2 ) + x 2 )u x) 2, δ δ > 2 + δ 2 ) 2 + x 2 )Ωh ; δ)u x) 2, u x < δ; 2 + δ 2 ) 2 + x 2 u x)4 ) ; δ)u x) 2, u x δ. δ 4 Ωh 2 + δ 2 ) 2 + x 2 )Ωh ; δ) + u x)4 δ 4 ) u x) 2. Now, selecting δ <, from 3.), we obtain Λu, x) 8 + x 2 )Ωh ; δ) u x) 2 + u x)2 u x) 4 ) 3.2) Applying operator M η α,ρ on above inequality and considering Lemma 3, we obtain αm η α,ρλu, x); x) 8α + x 2 )Ωh ; δ) M η α,ρu x) 2 ; x) ) + δ 4 Mη α,ρu x) 6 ; x) ) = 8α + x 2 )Ωh ; δ) O + δ )) α α, 3.3) as α. Now, choosing δ = α, we obtain δ 4 3.4) αm η α,ρλu, x); x) 8α + x 2 )Ω h ; )O). α On collecting 3.), 3.4)and using Lemma 2, we arrive to required result. In our next result we discuss Grüss-Voronovskaja type theorem for the operator defined by.3). Grüss inequality [2] measures the difference of integral of two functions with the product of integral of the two functions. Acu et al. [4], were first to show the application of Grüss inequality in approximation theory. In [], Gonska and Tachev discussed Grüss-type inequality using second order modulus of smoothness. Gal and Gonska [], proved Grüss-Voronovskaya estimates for the first time using Grüss inequality for Bernstein operators and for a class of

27 Miheşan-Kantorovich operators of blending type 25 Bernstein-Durrmeyer polynomials of real and complex variables. Recently, Tariboon and Ntouyas [23] introduced Grüss inequality in q-calculus. After that in [24] and [] authors investigated q-grüss-voronovskaja theorem for q-baskakov operators and q-szász operators respectively. In [2], Acar et al. discussed new forms of Voronovskaja type theorem in weighted spaces. Very recently, Deniz [7], obtained Grüss-Voronovskaja theorem for Jain-Kantorovich operators. Theorem 9 For h, g, h, g, h, g, hg), hg) C k σr + ), we have the following equality lim α α{mη α,ρhg; x) M η α,ρh; x)m η α,ρg; x)} = xlx + )h x)g x). Proof. By a straightforward calculation, we may write α{m η α,ρhg; x) M η α,ρh; x)m η α,ρg; x)} = α { M η α,ρhg; x) hx)gx) M η α,ρu x; x)hg) x) Mη α,ρu x) 2 [ ; x) hg) x) gx) M η 2! α,ρh; x) hx) M η α,ρu x; x)h x) Mη α,ρu x) 2 ] ; x) h x) M η 2! α,ρh; x) [ M η α,ρg; x) gx) M η α,ρu x; x)g x) Mη α,ρu x) 2 ] ; x) g x) 2! +2 Mη α,ρu x) 2 ; x) h x)g x) + g x) Mη α,ρu x) 2 ; x) 2! 2! } [hx) M η α,ρh; x)]g x)m η α,ρu x); x)[hx) M η α,ρh; x)]. Now, in view of Theorem 6, it follows that M η α,ρh; x) hx), as α and using Theorem 8, we have α{m η α,ρh; x) hx) M η α,ρu x; x)h x) Mη α,ρu x) 2 ; x) h x)}, as α, 2! since h, h C k σr + ). Thus, using Theorem 6, 8 and Lemma 3, we obtain the required result lim α α{mη α,ρhg; x) M η α,ρh; x)m η α,ρg; x)} = xlx + )h x)g x). Acknowledgement: The first author expresses her sincere thanks to The Ministry of Human Resource and Development, India for the financial assistance without which the above work would not have been possible.

28 26 S. Deshwal, P.N. Agrawal References [] T. Acar, Quantitative q-voronovskaya and q-grüss-voronovskaya-type results for q-szász Operators, Georgian Math. J., vol. 23, no. 4, 26, [2] T. Acar, A. Aral, I. Rasa, The new forms of Voronovskaja s theorem in weighted spaces, Positivity, vol. 2, 26, [3] T. Acar and G. Ulusoy, Approximation by modified Szász-Durrmeyer operators, Period. Math. Hung., vol. 72, no., 26, [4] A. M. Acu, H. Gonska, I. Rasa, Grüss-type and Ostrowski-type inequalities in approximation theory, Ukrainian Math. J, vol. 63, no. 6, 2, [5] P. N. Agrawal, V. Gupta, A. Satish, A. Kajla, Generalized Baskakov- Sz asz type operators, Appl. Math. Comput., vol. 236, 24, [6] C. Atakut, N. Ispir, Approximation by modified Szász-Mirakjan operators on weighted spaces, Proc. Indian Acad. Sci. Math., vol. 2, 22, [7] E. Deniz, Quantitative estimates for Jain-Kantorovich operators, Commun. Fac. Sci. Univ. Ank. Sr. A Math. Stat., vol. 65, 26, [8] Z. Ditzian, V. Totik, Moduli of smoothness, Springer-Verlag, New York, 987. [9] A.D. Gadzhiev, The convergence problem for a sequence of positive linear operators on unbounded sets and theorem analogue to that of P.P. Korovkin, Soviet Math. Dokl., vol. 5, no. 5, 974, [] S. G. Gal, H. Gonska, Grüss and Grüss-Voronovskaya-type esstimates for some Bernstein-type polynomials of real and complex variables, Jaen J. Approx., vol. 7, 25, [] H. Gonska and G. Tachev, Grüss type inequality for positive linear operators with second order moduli, Mat. vesnik, vol. 63, no. 4, 2, b [2] G. Grüss, Über das Maximum des absoluten Betrages von b a a fx)gx)dx b b a) 2 a fx)dx. b a gx)dx, Math Z., vol. 39, 935, [3] V. Gupta, p,q)- Szász- Mirakyan- Baskakov Operators, Complex Anal. Oper. Theory, 25, DOI:.7/s [4] V. Gupta, Direct estimates for new general family of Durrmeyer type operators, Boll. Unione Mat. Ital., vol. 7, no. 4, 25, [5] V. Gupta, T. M. Russias, Direct estimates for certain Szász type operators, Appl. Math. Comput., vol. 25, 25,

29 Miheşan-Kantorovich operators of blending type 27 [6] A. Kajla, T. Acar, A new modification of Durrmeyer type mixed hybrid operators, Carpathian J. Math., vol. 33, no. 3, 27, -9. [7] A. Kajla, A. M. Acu, P. N. Agrawal, Baskakov-Szász type operators based on inverse Polya-Eggenberger distribution, Ann. Funct. Anal., vol. 8, 27, [8] A. Kajla, S. Araci, Blending type approximation by Stancu-Kantorovich operators based on Polya-Eggenberger distribution, Open Phy., vol. 5, 27, [9] P. P. Korovkin, On convergence of linear positive operators in the spaces of continuous functions Russian), Doklady Akad. Nauk. SSSR NS), vol. 9, 953, [2] V. Miheşan, Gamma approximating operators, Creative Math. Inf., vol. 7, 28, [2] M. A. Ozarslan, O. Duman, Smoothness properties of modified Bernstein- Kantorovich operators, Numer. Funct. Anal. Optim., vol. 37, 26, [22] H. M. Srivastava, V. Gupta, A certain family of summation integral-type operators, Math. Comput. modelling, vol. 37, 23, [23] J. Taiboon, K. N. Ntouyas, Quantum integral inequalities on finite intervals, J. Inequal. Appl., 24, Article ID 2. [24] G. Ulusoy, T. Acar, q-voronovskaya type theorems for q-baskakov operators, Math. Methods Appl. Sci., 25,DOI.2/mma [25] S. Varma, F. Taşdelen, Szász type operators involving Charlier polynomials, Math. Comput. modelling, vol. 56, 22, [26] I. Yüksel, N. Ispir, Weighted approximation by a certain family of summation integral-type operators, Commut. Math. Appl., vol. 52, 26, Sheetal Deshwal Indian Institute of Technology Roorkee, India Research Scholar Department of Mathematcis IIT Roorkee, Uttarakhand , India shetald99@gmail.com P. N. Agrawal Indian Institute of Technology Roorkee, India Faculty Department of Mathematics IIT Roorkee, Uttarakhand , India pnappfma@gmail.com

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31 General Mathematics Vol. 25, No ), Some families of rational Heun functions and combinatorial identities Bărar Adina - Elena Abstract Continuing some previous investigations, we present new families of rational and polynomial Heun functions, and new combinatorial identities. 2 Mathematics Subject Classification: 33E3, 5A9. Key words and phrases: Heun functions, combinatorial identities Introduction The general Heun equation is where and y x) + γ x + δ x + ɛ αβx q x a )y x) + )yx) =, xx )x a) a / {, }, γ / {,, 2,...} α + β + = γ + δ + ɛ. Its solution yx) normalized by y)= is called the local) Heun function and is denoted by Hla, q; α, β; γ, δ; x). See,e.g., [],[8],[7] and the references therein. It was proved in [9] that Hl n, n; 2n, ;, ; x) = 2 k= k= ) n x k x) n k ) 2, k Hl 2, n; 2n, ;, ; x) = ) n + k x k + x) n k ) 2. k Received August, 27 Accepted for publication in revised form) October, 27 29

32 3 A. E. Bărar These functions are related to the Rényi entropy and Tsallis entropy; see, e.g., [9] and [3]. Closed forms of the functions Hl, 2nθ; 2n, 2θ; γ, γ; x) 2 and Hl, 2nθ; 2n, 2θ; γ, γ; x) 2 were presented in [2] and []. The corresponding results were used in order to derive several combinatorial identities, generalizing some formulas from [4]. Closed forms of Heun functionsin particular, rational and polynomial Heun functions) are important in applications; see, e.g., [7], [5],[] and the references therein. In this paper we continue the investigations from []. We present new families of rational and polynomial Heun functions, and new combinatorial identities. We shall use the notation a) :=, a) k := aa + )...a + k ), k, a nk := 4 n 2k k 2 Heun functions ) ) 2n 2k, r nk := n k In what follows we need the following results. ) n a nk. k Theorem [6],[2,Prop], [2;4)],[2;5)],[]) Let αβ. Then ) Hl 2, α + 2)β + 2); α + 2, β + 2; γ +, γ + ; x) = 2 = γ d 2x) αβ dx Hl 2, αβ; α, β; γ, γ; x), 2 2) Hl 2, 2γ α)2γ β); 2γ α, 2γ β; γ +, γ + ; x) = 2 = γ d 2x)α+β+ 2γ αβ dx Hl 2, αβ; α, β; γ, γ; x). 2 Theorem 2 [2;Th.],[2;Cor.2]) i)let n be an integer and θ R. Then 3) Hl n ) n 2, 2nθ; 2n, 2θ; γ, γ; x) = 4 k θ)k x 2 x) k. k γ) k k=

33 Some families of rational Heun functions and combinatorial identities 3 ii)let γ and n be integers, < γ n, and θ R. Then 4) Hl, 2nθ; 2n, 2θ; γ, γ; x) = 2 n γ = 2x) 2γ n θ) k= Now we can state the main results of this section. Theorem 3 Let m n. Then ) n γ γ 4 k θ)k x 2 x) k. k γ) k 5) Hl, n + ; 2n + 2, ; m + 2, m + 2; x) = 2 n m k= m + )22m+ = n m)2m + ) m + k + k + ) m n m = 2x) 2m 2n+ k= Proof. According to [,Th2.] we have ) n ) 2m m m ) a n,m+k+ 2x) 2m 2n++2k ) n m m + 4 k 3 2 ) k x 2 x) k. k m + 2) k 6) Hl, 2m + )m n); 2m n), 2m + ; m +, m + ; x) = 2 ) n 2m = 4 m m m By using 2) and 6) we get ) n m ) m + j a n,m+j 2x) 2j. m j= Hl, n + ; 2n + 2, ; m + 2, m + 2; x) = 2 = m + 2m n)2m + ) 2x)2m 2n d dx Hl, 2m + )m n); 2m n), 2m + ; m +, m + ; x) = 2 ) m + n ) 2m = 2m n)2m + ) 2x)2m 2n 4 m m m

34 32 A. E. Bărar n m j= m + j m ) a n,m+j 4j) 2x) 2j. Now the first equality in 5) follows by setting j=k+. In order to prove that the first member and the last member of 5) are equal, it suffices to use 4) with γ = m + 2, θ = 2 and n replaced by n+. Remark Since 2m 2n +, 5) shows that Hl 2, n + ; 2n + 2, ; m + 2, m + 2; x) is a rational function for which 2 order 2n 2m. In particular, for m = n we get is a pole of Hl, n + ; 2n + 2, ; n +, n + ; x) = 2 2x. Theorem 4 Let k n. Then 7) Hl, k n + )2k + 3); 2k n + ), 2k + 3; 2k + 2, 2k + 2; x) = 2 n k j= = 22k+ n k n + k n 2n 2j 2 j + ) 2k = n k j= Proof. From [;2.2)] we know that ) ) n k ) n j + ) r n,k+j+ 2x) 2j = ) n k k + 4 j 3 2 ) j x 2 x) j. j 2k + 2) j 8) Hl, k n)2k + ); 2k n), 2k + ; 2k +, 2k + ; x) = 2 = 4 k n + k n ) n k Combining ) and 8) we get ) n k 2n 2i 2k i= ) n i ) r n,k+i 2x) 2i. Hl, k n + )2k + 3); 2k n + ), 2k + 3; 2k + 2, 2k + 2; x) = 2 = d 2x) 2k n) dx Hl, k n)2k + ); 2k n), 2k + ; 2k +, 2k + ; x) = 2

35 Some families of rational Heun functions and combinatorial identities 33 = 22k+ n k n + k k ) n k ) n k 2n 2i i 2k i= ) n i ) r n,k+i 2x) 2i 2. In order to prove the first equality in 7), it suffices to replace i by j+. The second equality is a consequence of 3), if we put γ = 2k + 2, θ = k + 3 2, and replace n by n-k-. Remark 2 From Theorem 4 we see that Hl 2, k n + )2k + 3); 2k n + ), 2k + 3; 2k + 2, 2k + 2; x) is a polynomial of degree 2n-k-). In particular, Hl 2n, 2n; 2, 2n ; 2n 2, 2n 2; x) = + 2 n x2 x). Theorem 5 Let k n. Then 9) Hl, 2k + 3)k n + ); 2k n + ), 2k + 3; 2k + 3, 2k + 3; x) = 2 n k j= = 4k+ n + n + k + n ) ) n k + ) ) 2k + 2j + 2 n + r nj n k j) 2x) 2n 2k 2j 2 = 2j k + j + = n k j= Proof. According to[;2.2)] we have ) n k k + 4 j 3 2 ) j x 2 x) j. j 2k + 3) j ) Hl, k n)2k + ); 2k n), 2k + ; 2k + 2, 2k + 2; x) = 2 = 4 k 2k + ) n + k + ) n n + n k n k ) ) 2k + 2j + 2 n + r nj 2x) 2n 2k 2j. 2j k + j + j= By using ) and ) it follows that Hl, 2k + 3)k n + ); 2k n + ), 2k + 3; 2k + 3, 2k + 3; x) = 2 = 2k + 2 2k n)2k + ) 2x)

36 34 A. E. Bărar j= d dx Hl, 2k + )k n); 2k n), 2k + ; 2k + 2, 2k + 2; x) = 2 = 2k + 2 2k + 4k 2k n)2k + ) n + n + k + n ) ) n k n k ) ) 2k + 2j + 2 n + r nj 4)n k j) 2x) 2n 2k 2j 2 = 2j k + j + n k j= = 4k+ k + ) n k)n + ) n + k + n ) ) n k ) ) 2k + 2j + 2 n + r nj n k j) 2x) 2n 2k 2j 2. 2j k + j + This leads immediately to the first equality in 9). The second equality in 9) is a consequence of 3) with γ = 2k + 3, θ = k and n replaced byn-k-. Remark 3 Theorem 5 shows that Hl 2, 2k+3)k n+); 2k n+), 2k+3; 2k+3, 2k+3; x) is a polynomial of degree 2n-k-). For exemple, Hl 2, 2n; 2, 2n ; 2n, 2n ; x) = 2x2 2x +. 3 Combinatorial identities By using the second equality in 5) we get = n m = j= n m k= n m = k= m + )2 2m+ ) n ) 2m n m)2m + ) m m n m j= m + k + k + ) m ) a n,m+k+ 2x) 2k = ) n m m + 4 j 3 2 ) j x 2 x) j = j m + 2) j ) n m m + 4 j 3 2 ) j 4 j j m + 2) j n m j=k From this we derive ) n m m + 3 j j k= 2 ) j m + 2) j ) j ) j k 2x) 2k = k ) j ) j k 2x) 2k. k

37 Some families of rational Heun functions and combinatorial identities 35 Corollary Let m n and k n m. Then n m j=k ) n m m + 3 j 2 ) j m + 2) j ) = 22m+ 2m ) n k + ) 2m + m m + ) j ) j k = k m + k + m ) a n,m+k+. Exemple In the preceding corollary set m = k = and replace n by n+; we get n ) ) ) n 2j + 2 4) j 2n = j j + n + )2 2n n j= Coming back to the second equality in 5), we can write n m n m k= n m j= = j= ) n m m + 4 k 3 2 ) k x 2 x) k = k m + 2) k ) = 22m+ 2m ) n 2m + m m + m + j + j + ) m ) a n,m+j+ 2x) 2j = ) = 22m+ 2m ) n 2m + m m + m + j + j + ) m n m k= This leads to the following n m j=k 2 2m+ 2m + ) a n,m+j+ j k= ) j 4 k x 2 x) k = k ) 2m ) n m m + ) ) m + j + j j + ) 4 k a n,m+j+ x 2 x) k. m k Corollary 2 Let m n and k n m. Then n m j=k ) m + j + j j + ) m k ) a n,m+j+ = 2m + 2 2m+ ) ) n n m m ) k. m + k m + 2) k ) 2m m

38 36 A. E. Bărar Exemple 2 For m=k= and n replaced by n+, the previous identity reduces to n ) ) 2n 2j 2j + 2 j + ) = n + )2 2n+. n j j + j= See also [;2.9)]. References [] A. Bărar, G. Mocanu, I. Raşa, Heun functions and combinatorial identities, Preprint, 27. [2] A. Bărar, G. Mocanu, I. Raşa, Heun functions related to entropy, 27. [3] A. Bărar, G. Mocanu, I. Raşa, Bounds for some entropies and special functions, to appear in Carpathian J.Math., 28. [4] H. W. Gould, Combinatorial Identities, Morgantown, W.Va., 972. [5] N. Gurappa, P. K. Panigrahi, On polynomial solutins of Heun equation, J. Phys. A: Math. Gen., no. 37, 24, [6] A. Ishkhanyan, K. A. Souminem, New solutions of Heun s general equation, J. Phys. A: Math. Gen., no. 36, 23, [7] G. Kristensson, Second order Differential Equations. Special Functions and Their Classification, Springer, 2. [8] R. S. Maier, The 92 solutions of the Heun equation, Math. Comp., no. 76, 27, [9] I. Raşa, Entropies and Heun functions associated with positive linear operators, Appl. Math Comput., no. 268, 25, [] Ronveaux A, editor. Heun s Differential Equations, London: Oxford University Press, 995. [] Y.-Zh. Zhang, Exact polynomial solutions of second order differential equations and their applications, arxiv: 7.59 v[math-ph], 2. Bărar Adina - Elena Technical University of Cluj-Napoca Department of Mathematics Str Memorandumului nr 28, 44 Cluj-Napoca, Romania ellena sontica@yahoo.com

39 General Mathematics Vol. 25, No ), Voronovskaja-type formulas and applications Fadel Nasaireh Abstract We continue previous investigations concerning Voronovskaja-type formulas for inverses of positive linear operators. 2 Mathematics Subject Classification: 4A36. Key words and phrases: positive operators; nonpositive operators; Voronovskaja type formulas. Introduction In this paper we continue previous investigations see [5], [6], [8]) concerning Voronovskaja-type formulas for inverses of classical positive linear operators. Section 2 is devoted to Voronovskaja-type formulas for Un, where U n is the genuine Bernstein-Durrmeyer operator. In Section 3 we present an application of the Voronovskaja-type formula for B n, the inverse of the Beta operator of Mühlbach and Lupaş. Section 4 contains an application involving Bernstein operators. Throughout the paper we consider the Banach space C [, ] of all continuous, real-valued functions, equipped with the supremum norm. For a function f C[, ], n N, x [, ] the Beta operators are given by f), x =, B n fx) = Bnx,n nx) t nx t) n x) ft)dt, < x <, f), x =, with Euler s Beta function Bx, y) = tx t) y dt, x, y >. The classical Bernstein operators B n : C[, ] C[, ] are defined by B n f; x) = n p n,j x)f j= j n ), x [, ], Received June, 27 Accepted for publication in revised form) 8 September, 27 37

40 38 F. Nasaireh where p n,j x) = ) n x j x) n j. j The genuine Bernstein-Durrmeyer operators U n : C[, ] Π n, are explicitly given by U n f; x) = f)p n, x) + f)p n,n x)+ n n ) p n,k x) k= p n 2,k t)ft)dt. where p,k = and p,k =. By Π we denote the set of all polynomial functions defined on [, ]. Π m stands for the subset of Π consisting of the polynomial functions of degree m. 2 Voronovskaja-type formulas for U n We recall a result from [8]; see also [6]. Let X be a Banach space and W Z Y linear subspaces of X. Let A, B : Y X; U, V : Z X; S, T : W X be linear operators. Consider also two sequences of linear operators P n : X X, Q n : Y X, n, and suppose that each P n is bounded. Theorem [8; Th.2.]) i) Suppose that ) lim n P nx = x, x X, 2) lim n np ny y) = Ay ; Then lim nq ny y) = By, y Y. n 3) lim n np nq n y y) = Ay + By, y Y. ii) In addition to ) and 2), suppose that 4) Bz Y, z Z, 5) [ lim n n ] np n z z) Az ] [ lim n nq n z z) Bz n = Uz; = V z, z Z. Then [ ] 6) lim n np n Q n z z) Az Bz = Uz + V z + ABz, z Z. n

41 Voronovskaja-type formulas and applications 39 iii) Let ), 2), 4), 5) be satisfied. Moreover, suppose that for each w W we have V w Y, Bw Z, and { [ ] } lim n n np n w w) Aw Uw = Sw, n { [ ] } lim n n nq n w w) Bw V w = T w. n Then, for all w W, { [ ] lim n n np n Q n w w) Aw Bw n Now we are in a position to prove Sw + T w + AV w + UBw. } Uw V w ABw = Theorem 2 Let m and p n Π m, n. Suppose that the sequence p n ) is uniformly convergent on [, ] to p Π m. Then 7) lim n n U n p n t) p n t) ) = t t) p 2) t), 8) uniformly on [, ]. [ lim n n U n n p n t) p n t) ) ] + t t) p 2) t) = t t) [ ] t t) p 4) t) + 2 2t) p 3) t) 2p 2) t), 2 Proof. It is well-known see, e.g., [], [2], [4], [7], [9], []) that U n has eigenpolynomials q t) =, q t) = t, with eigenvalue, and with eigenvalues λ nk = q k t) = dk 2 dt k 2 [ t k t) k ], k = 2,..., n, n )!n! n +k)!n k)!. Moreover, 9) t t) q 2) k t) + k k ) q k t) =, k. 7) Let n k. Then U n q k = λ nk q k, and therefore U n q k = λ nk q k. For t [, ] we have lim n U n n q k t) q k t) ) = lim n )q k t) = n λ ) nk n n + )... n + k ) lim n n n n )... n k + ) q k t) = k k ) q k t). Thus, according to 9), we have lim n n U n q k t) q k t) ) = t t) q 2) k t), k,

42 4 F. Nasaireh uniformly on [, ]. This leads immediately to ) lim n n U n pt) pt) ) = t t) p 2) t), p Π, uniformly on [, ]. Now let p n = m j= a nje j Π m be uniformly convergent on [, ] to p = m j= a je j Π m. Then lim a nj = a j, and ) leads to n lim n U n n p n t) p n t) ) = lim m n j= a nj n U n e j t) e j t) ) = m ) 2) = t t) a j e j t) = t t) p 2) t). j= This proves 7). 8) Let again n k and t [, ]. Then [ n n Un q k t) q k t) ) ] + t t) q 2) k t) = [ ) ] n n + )... n + k ) = n n n n )... n k + ) q k t) + t t) q 2) k t) = [ n k ] k k ) + terms of degree < k = n k k ) q k t). n ) n 2)... n k + ) Consider the operators P n := U n, Q n := U n, Up t) = Ap t) := t t) p 2) t), Bp t) := t t) p 2) t), [ t t) p 4) t) + 2 2t) p 3) t) 2p 2) t) t t) 2 According to [8; Example 2.2], we have Up = lim n n [n P np p) Ap], p Π. The above computation shows that there exists the operator [ ] ) V p t) := lim n n Q n p t) p t)) Bp t), p Π. n According to 6), 2) Up + V p + ABp =, p Π. 3) By a straightforward calculation, 2) yields t t) [ V p t) = t t) p 4) t) + 2 ] +2 2t) p 3) t) 2p 2) t), p Π. ].