A Kantorovich Type of Szasz Operators Including Brenke-Type Polynomials

Abstract

We give a Kantorovich variant of a generalization of Szasz operators defined by means of the Brenke-type polynomials and obtain convergence properties of these operators by using Korovkin's theorem. We also present the order of convergence with the help of a classical approach, the second modulus of continuity, and Peetre's 𝐾-functional. Furthermore, an example of Kantorovich type of the operators including Gould-Hopper polynomials is presented and Voronovskaya-type result is given for these operators including Gould-Hopper polynomials.

1. Introduction

The Szasz operators (also called Szasz-Mirakyan operators) which are defined by [1] 𝑆_𝑛𝑓;𝑥∶=𝑒^{∞−𝑛𝑥}_𝑘=0(𝑛𝑥)^𝑘𝑓𝑘𝑘!𝑛, (1.1) where 𝑛∈ℕ, 𝑥≥0, and 𝑓∈𝐶[0,∞) have an important role in the approximation theory, and their approximation properties have been investigated by many researchers.

In [2], Jakimovski and Leviatan proposed a generalization of Szasz operators by means of the Appell polynomials 𝑝_𝑘(𝑥) which have the generating functions of the form: g(𝑡)𝑒^𝑡𝑥=^∞_𝑘=0𝑝_𝑘(𝑥)𝑡^𝑘, (1.2) where g(𝑧)=∑^∞_𝑘=0𝑎_𝑘𝑧^𝑘(𝑎₀≠0) is an analytic function in the disc |𝑧|<𝑅, (𝑅>1) and g(1)≠0. Under the assumption that 𝑝_𝑘(𝑥)≥0 for 𝑥∈[0,∞), Jakimovski and Leviatan [2], defined the following linear positive operators: 𝑃_𝑛𝑒𝑓;𝑥∶=^−𝑛𝑥g(1)^∞_𝑘=0𝑝_𝑘()𝑓𝑘𝑛𝑥𝑛. (1.3)

After that, Ismail [3] defined another generalization of Szasz operators involving the operators (1.1) and (1.3) by means of Sheffer polynomials. Let 𝐴(𝑧)=∑^∞_𝑘=0𝑎_𝑘𝑧^𝑘(𝑎₀≠0) and 𝐻(𝑧)=∑^∞_𝑘=1ℎ_𝑘𝑧^𝑘(ℎ₁≠0) be analytic functions in the disc |𝑧|<𝑅,(𝑅>1). Here, 𝑎_𝑘 and ℎ_𝑘 are real. The Sheffer polynomials 𝑝_𝑘(𝑥) are generated by 𝐴(𝑡)𝑒^{𝑥𝐻(𝑡)}=^∞_𝑘=0𝑝_𝑘(𝑥)𝑡^𝑘. (1.4) With the help of these polynomials, Ismail constructed the following linear positive operators: 𝑇_𝑛𝑒𝑓;𝑥∶=^{−𝑛𝑥𝐻(1)}𝐴(1)^∞_𝑘=0𝑝_𝑘()𝑓𝑘𝑛𝑥𝑛,𝑛∈ℕ (1.5) under the assumptions(i)for 𝑥∈[0,∞), 𝑝_𝑘(𝑥)≥0,(ii)𝐴(1)≠0 and 𝐻^′(1)=1.

Later, Varma et al. [4] defined another generalization of Szasz operators by means of the Brenke-type polynomials. Suppose that 𝐴(𝑡)=^∞_𝑟=0𝑎_𝑟𝑡^𝑟,𝑎₀≠0,𝐵(𝑡)=^∞_𝑟=0𝑏_𝑟𝑡^𝑟,𝑏_𝑟≠0(𝑟≥0) (1.6) are analytic functions. The Brenke-type polynomials [5] have generating functions of the form 𝐴(𝑡)𝐵(𝑥𝑡)=^∞_𝑘=0𝑝_𝑘(𝑥)𝑡^𝑘 (1.7) from which the explicit form of 𝑝_𝑘(𝑥) is as follows: 𝑝_𝑘(𝑥)=^𝑘_𝑟=0𝑎_𝑘−𝑟𝑏_𝑟𝑥^𝑟,𝑘=0,1,2,…. (1.8)

Under the assumptions (i)𝐴(1)𝑎≠0,^𝑘−𝑟𝑏_𝑟𝐴(1)()[)⟶(),()()|𝑡|(),≥0,0≤𝑟≤𝑘,𝑘=0,1,2,…,ii𝐵∶0,∞0,∞iii)(1.6and1.7convergefor<𝑅,𝑅>1 (1.9)

Varma et al. introduced the linear positive operators 𝐿_𝑛(𝑓;𝑥) via 𝐿_𝑛1𝑓;𝑥∶=𝐴(1)𝐵(𝑛𝑥)^∞_𝑘=0𝑝_𝑘()𝑓𝑘𝑛𝑥𝑛, (1.10) where 𝑥≥0 and 𝑛∈ℕ.

The aim of this paper is to present a Kantorovich type of the operators given by (1.10) and to give their some approximation properties. We consider the Kantorovich version of the operators (1.10) under the assumptions (1.9) as follows: 𝐾_𝑛𝑛𝑓;𝑥∶=𝐴(1)𝐵(𝑛𝑥)^∞_𝑘=0𝑝_𝑘(𝑛𝑥)^{(𝑘+1)/𝑛}_𝑘/𝑛𝑓(𝑡)𝑑𝑡, (1.11) where 𝑛∈ℕ, 𝑥≥0, and 𝑓∈𝐶[0,∞). It is easy to see that 𝐾_𝑛 defined by (1.11) is linear and positive.

In the case of 𝐵(𝑡)=𝑒^𝑡 and 𝐴(𝑡)=1, with the help of (1.7), it follows that 𝑝_𝑘(𝑥)=𝑥^𝑘/𝑘!, so the operators (1.11) reduce to the Szasz-Mirakyan-Kantorovich operators defined by [6] 𝐾_𝑛𝑓;𝑥∶=𝑛𝑒^{∞−𝑛𝑥}_𝑘=0(𝑛𝑥)^𝑘𝑘!^{(𝑘+1)/𝑛}_𝑘/𝑛𝑓(𝑡)𝑑𝑡. (1.12) Various approximation properties of the Szasz-Mirakyan-Kantorovich operators and their iterates may be found in [7–13].

The case of 𝐵(𝑡)=𝑒^𝑡 gives the Kantorovich version of the operators (1.3).

The structure of the paper is as follows. In Section 2, the convergence of the operators (1.11) is given by means of Korovkin's theorem. The order of approximation is obtained with the help of a classical approach, the second modulus of continuity, and Peetre's 𝐾-functional in Section 3. Finally, as an example, we present a Kantorovich type of the operators including Gould-Hopper polynomials and then we give a Voronovskaya-type theorem for the operators including Gould-Hopper polynomials.

2. Approximation Properties of 𝐾_𝑛 Operators

In this section, we give our main theorem with the help of Korovkin theorem. We begin with the following lemma which is necessary to prove the main result.

Lemma 2.1. For all 𝑥∈[0,∞), the operators 𝐾_𝑛 defined by (1.11) verify 𝐾_𝑛()𝐾1;𝑥=1,_𝑛()=𝐵𝑠;𝑥^′(𝑛𝑥)𝐵(𝑛𝑥)𝑥+2𝐴^′(1)+𝐴(1)(1),𝐾2𝑛𝐴_𝑛𝑠²=𝐵;𝑥^′′(𝑛𝑥)𝐵()𝑥𝑛𝑥²+2𝐵^′(𝑛𝑥)𝐴^′(1)+𝐴(1)(1)𝐵()1𝑛𝐴𝑛𝑥𝑥+𝑛²𝐴(1)𝐴^′′(1)+2𝐴^′(1)+𝐴(1)3. (2.1)(2.2)(2.3)

Proof. Using the generating function of the Brenke-type polynomials given by (1.7), we can write ^∞_𝑘=0𝑝_𝑘(𝑛𝑥)=𝐴(1)𝐵(𝑛𝑥_∞𝑘=0),𝑘𝑝_𝑘(𝑛𝑥)=𝐴^′(1)𝐵(𝑛𝑥)+𝑛𝑥𝐴(1)𝐵^′(𝑛𝑥_∞𝑘=0),𝑘²𝑝_𝑘(𝑛𝑥)=𝑛²𝑥²𝐴(1)𝐵^′′(𝑛𝑥)+𝑛𝑥𝐵^′(𝑛𝑥)2𝐴^′(1)+𝐴(1)+𝐵(𝑛𝑥)𝐴^′′(1)+𝐴^′(1). (2.4) From these equalities, the assertions of the lemma are obtained.

Lemma 2.2. For 𝑥∈[0,∞), one has 𝐾_𝑛(𝑠−𝑥)²=𝐵;𝑥^′′(𝑛𝑥)−2𝐵^′(𝑛𝑥)+𝐵(𝑛𝑥)𝐵()𝑥𝑛𝑥²+2𝐴^′(1)𝐵^′(𝑛𝑥)−𝐵(𝑛𝑥)+𝐴(1)2𝐵^′(𝑛𝑥)−𝐵(𝑛𝑥)(1)𝐵()𝑥+1𝑛𝐴𝑛𝑥𝑛²𝐴(1)𝐴^′′(1)+2𝐴^′(1)+𝐴(1)3. (2.5)

Proof. From the linearity of 𝐾_𝑛, we get 𝐾_𝑛(𝑠−𝑥)²;𝑥=𝐾_𝑛𝑠²;𝑥−2𝑥𝐾_𝑛(𝑠;𝑥)+𝑥²𝐾_𝑛(1;𝑥). (2.6) Next, we apply Lemma 2.1.

Theorem 2.3. Let [),𝑓(𝑥)𝐸∶=𝑓∶𝑥∈0,∞1+𝑥²isconvergentas𝑥→∞,lim_𝑦→∞𝐵^′𝑦𝐵𝑦=1,lim_𝑦→∞𝐵^′′𝑦𝐵𝑦=1. (2.7)(2.8) If 𝑓∈𝐶[0,∞)∩𝐸, then lim_𝑛→∞𝐾_𝑛𝑓;𝑥=𝑓(𝑥), (2.9) and the operators 𝐾_𝑛 converge uniformly in each compact subset of [0,∞).

Proof. Using Lemma 2.1 and taking into account the equality (2.8) we get lim_𝑛→∞𝐾_𝑛𝑠^𝑖;𝑥=𝑥^𝑖,𝑖=0,1,2. (2.10) The above convergence is satisfied uniformly in each compact subset of [0,∞). We can then apply the universal Korovkin-type property (vi) of Theorem  4.1.4 in [14] to obtain the desired result.

3. The Order of Approximation

In this section, we deal with the rates of convergence of the 𝐾_𝑛(𝑓) to 𝑓 by means of a classical approach, the second modulus of continuity, and Peetre's 𝐾-functional.

Let 𝑓∈𝐶[0,∞). If 𝛿>0, the modulus of continuity of 𝑓 is defined by 𝑤𝑓;𝛿∶=sup_{[)𝑥,𝑦∈0,∞|𝑥−𝑦|≤𝛿}𝑓(𝑥)−𝑓𝑦, (3.1) where 𝐶[0,∞) denotes the space of uniformly continuous functions on [0,∞). It is also well known that, for any 𝛿>0 and each 𝑥∈[0,∞), 𝑓(𝑥)−𝑓𝑦≤𝑤𝑓;𝛿𝑥−𝑦𝛿+1. (3.2)

The next result gives the rate of convergence of the sequence 𝐾_𝑛(𝑓) to 𝑓 by means of the modulus of continuity.

Theorem 3.1. Let 𝑓∈𝐶[0,∞)∩𝐸. The 𝐾_𝑛 operators satisfy the following inequality: 𝐾_𝑛𝑓;𝑥−𝑓(𝑥)≤2𝑤𝑓;𝜆_𝑛(𝑥), (3.3) where 𝜆∶=𝜆_𝑛(𝑥)=𝐾_𝑛(𝑠−𝑥)²=𝐵;𝑥^′′(𝑛𝑥)−2𝐵^′(𝑛𝑥)+𝐵(𝑛𝑥)𝐵()𝑥𝑛𝑥²+2𝐴^′(1)𝐵^′(𝑛𝑥)−𝐵(𝑛𝑥)+𝐴(1)2𝐵^′(𝑛𝑥)−𝐵(𝑛𝑥)(1)𝐵()𝑥+1𝑛𝐴𝑛𝑥𝑛²𝐴(1)𝐴^′′(1)+2𝐴^′(1)+𝐴(1)3. (3.4)

Proof. Using (2.1), (3.2), and the linearity property of 𝐾_𝑛 operators, we can write 𝐾_𝑛(𝑥≤𝑛𝑓;𝑥−𝑓)𝐴(1)𝐵(𝑛𝑥)^∞_𝑘=0𝑃_𝑘(𝑛𝑥)^{(𝑘+1)/𝑛}_𝑘/𝑛𝑓(𝑠)(𝑥≤𝑛−𝑓)𝑑𝑠𝐴(1)𝐵(𝑛𝑥)^∞_𝑘=0𝑃_𝑘(𝑛𝑥)^{(𝑘+1)/𝑛}_𝑘/𝑛|𝑠−𝑥|𝛿𝑤≤𝑛+1𝑓;𝛿𝑑𝑠1+𝐴(1)𝐵(𝑛𝑥)𝛿^∞_𝑘=0𝑃_𝑘(𝑛𝑥)^{(𝑘+1)/𝑛}_𝑘/𝑛|𝑠−𝑥|𝑑𝑠𝑤𝑓;𝛿. (3.5) By using the Cauchy-Schwarz inequality for integration, we get ^{(𝑘+1)/𝑛}_𝑘/𝑛||1𝑠−𝑥𝑑𝑠≤√𝑛^{(𝑘+1)/𝑛}_𝑘/𝑛|𝑠−𝑥|²𝑑𝑠^1/2 (3.6) which holds that ^∞_𝑘=0𝑃_𝑘(𝑛𝑥)^{(𝑘+1)/𝑛}_𝑘/𝑛||1𝑠−𝑥𝑑𝑠≤√𝑛^∞_𝑘=0𝑃_𝑘(𝑛𝑥)^{(𝑘+1)/𝑛}_𝑘/𝑛|𝑠−𝑥|²𝑑𝑠^1/2. (3.7) By applying the Cauchy-Schwarz inequality for summation on the right-hand side of (3.7), we have ^∞_𝑘=0𝑃_𝑘(𝑛𝑥)^{(𝑘+1)/𝑛}_𝑘/𝑛|𝑠−𝑥|𝑑𝑠≤𝐴(1)𝐵(𝑛𝑥)√𝑛𝐴(1)𝐵(𝑛𝑥)𝑛𝐾_𝑛(𝑠−𝑥)²;𝑥^1/2=𝐴(1)𝐵(𝑛𝑥)𝑛𝐾_𝑛(𝑠−𝑥)²;𝑥^1/2=𝐴(1)𝐵(𝑛𝑥)𝑛𝜆_𝑛(𝑥)^1/2, (3.8) where 𝜆_𝑛(𝑥) is given by (3.4). If we use this in (3.5), we obtain 𝐾_𝑛(𝑥≤1𝑓;𝑥−𝑓)1+𝛿𝜆_𝑛(𝑥)𝑤𝑓;𝛿. (3.9) On choosing 𝛿=𝜆_𝑛(𝑥), we arrive at the desired result.

Recall that the second modulus of continuity of 𝑓∈𝐶_𝐵[0,∞) is defined by 𝑤₂𝑓;𝛿∶=sup_{0<𝑡≤𝛿}𝑓(⋅+2𝑡)−2𝑓(⋅+𝑡)+𝑓(⋅)_𝐶𝐵, (3.10) where 𝐶_𝐵[0,∞) is the class of real valued functions defined on [0,∞) which are bounded and uniformly continuous with the norm ‖𝑓‖_𝐶𝐵=sup_{𝑥∈[0,∞)}|𝑓(𝑥)|.

Peetre's 𝐾-functional of the function 𝑓∈𝐶_𝐵[0,∞) is defined by 𝐾𝑓;𝛿∶=inf_{𝑔∈𝐶2𝐵[)0,∞}𝑓−𝑔_𝐶𝐵+𝛿𝑔_𝐶2𝐵, (3.11) where 𝐶²_𝐵[0,∞)∶=𝑔∈𝐶_𝐵[0,∞)∶𝑔^′,𝑔^′′∈𝐶_𝐵[0,∞), (3.12) and the norm ‖𝑔‖_𝐶2𝐵∶=‖𝑔‖_𝐶𝐵+‖𝑔^′‖_𝐶𝐵+‖𝑔^′′‖_𝐶𝐵 (see [15]). It is clear that the following inequality: 𝐾𝑓;𝛿≤𝑀𝑤₂𝑓;√𝛿+min(1,𝛿)𝑓_𝐶𝐵, (3.13) holds for all 𝛿>0. The constant 𝑀 is independent of 𝑓 and 𝛿.

Theorem 3.2. Let 𝑓∈𝐶²_𝐵[0,∞). The following 𝐾_𝑛𝑓;𝑥−𝑓(𝑥)≤𝜁𝑓_𝐶2𝐵 (3.14) holds, where 𝜁∶=𝜁_𝑛(𝑥)=𝐵^′′(𝑛𝑥)−2𝐵^′(𝑛𝑥)+𝐵(𝑛𝑥)()𝑥2𝐵𝑛𝑥²+2𝐴^′(1)𝐵^′(𝑛𝑥)−𝐵(𝑛𝑥)+𝐴(1)2(𝑛+1)𝐵^′(𝑛𝑥)−(2𝑛+1)𝐵(𝑛𝑥)(1)𝐵()𝑥+12𝑛𝐴𝑛𝑥2𝑛²𝐴(1)𝐴^′′(1)+2𝐴^′(1)+𝐴(1)3+2𝐴^′(1)+𝐴(1)(1).2𝑛𝐴 (3.15)

Proof. From the Taylor expansion of 𝑓, the linearity of the operators 𝐾_𝑛 and (2.1), we have 𝐾_𝑛𝑓;𝑥−𝑓(𝑥)=𝑓^′(𝑥)𝐾_𝑛()+1𝑠−𝑥;𝑥2𝑓^′′𝜂𝐾_𝑛(𝑠−𝑥)²;𝑥,𝜂∈(𝑥,𝑠). (3.16) Since 𝐾_𝑛()=𝐵𝑠−𝑥;𝑥^′(𝑛𝑥)−𝐵(𝑛𝑥)𝐵(𝑛𝑥)𝑥+2𝐴^′(1)+𝐴(1)2𝑛𝐴(1)≥0 (3.17) for 𝑠≥𝑥, by considering Lemmas 2.1 and 2.2 in (3.16), we can write that 𝐾_𝑛(𝑥≤𝐵𝑓;𝑥−𝑓)^′(𝑛𝑥)−𝐵(𝑛𝑥)𝐵(𝑛𝑥)𝑥+2𝐴^′(1)+𝐴(1)(1)𝑓2𝑛𝐴^′𝐶_𝐵+12𝐵^′′(𝑛𝑥)−2𝐵^′(𝑛𝑥)+𝐵(𝑛𝑥)𝐵()𝑥𝑛𝑥²+2𝐴^′(1)𝐵^′(𝑛𝑥)−𝐵(𝑛𝑥)+𝐴(1)2𝐵^′(𝑛𝑥)−𝐵(𝑛𝑥)(1)𝐵()𝑥+1𝑛𝐴𝑛𝑥𝑛²𝐴(1)𝐴^′′(1)+2𝐴^′(1)+𝐴(1)3𝑓^′′_𝐶𝐵≤𝐵^′′(𝑛𝑥)−2𝐵^′(𝑛𝑥)+𝐵(𝑛𝑥)()𝑥2𝐵𝑛𝑥²+2𝐴^′(1)𝐵^′(𝑛𝑥)−𝐵(𝑛𝑥)+𝐴(1)2(𝑛+1)𝐵^′(𝑛𝑥)−(2𝑛+1)𝐵(𝑛𝑥)(1)𝐵()𝑥+12𝑛𝐴𝑛𝑥2𝑛²𝐴(1)𝐴^′′(1)+2𝐴^′(1)+𝐴(1)3+2𝐴^′(1)+𝐴(1)(1)𝑓2𝑛𝐴^𝐶2𝐵 (3.18) which completes the proof.

Theorem 3.3. Let 𝑓∈𝐶_𝐵[0,∞). Then 𝐾_𝑛𝑓;𝑥−𝑓(𝑥)≤2𝑀𝑤₂𝑓;√𝛿+min(1,𝛿)𝑓_𝐶𝐵, (3.19) where 𝛿∶=𝛿_𝑛(𝑥)=12𝜁_𝑛(𝑥) (3.20) and 𝑀>0 is a constant which is independent of the functions𝑓 and 𝛿. Also, 𝜁_𝑛(𝑥) is the same as in Theorem 3.2.

Proof. Suppose that 𝑔∈𝐶²_𝐵[0,∞). From Theorem 3.2, we can write 𝐾_𝑛𝑓;𝑥−𝑓(𝑥)≤𝐾_𝑛𝑓−𝑔;𝑥+𝐾_𝑛𝑔;𝑥−𝑔(𝑥)+𝑔(𝑥)−𝑓(𝑥)≤2𝑓−𝑔_𝐶𝐵+𝜁𝑔_𝐶2𝐵=2𝑓−𝑔_𝐶𝐵+𝛿𝑔_𝐶2𝐵. (3.21) The left-hand side of inequality (3.21) does not depend on the function 𝑔∈𝐶²_𝐵[0,∞), so 𝐾_𝑛𝑓;𝑥−𝑓(𝑥)≤2𝐾𝑓;𝛿, (3.22) where 𝐾(𝑓;𝛿) is Peetre's 𝐾-functional defined by (3.11). By the relation between Peetre's 𝐾-functional and the second modulus of smoothness given by (3.13), inequality (3.21) becomes 𝐾_𝑛𝑓;𝑥−𝑓(𝑥)≤2𝑀𝑤₂𝑓;√𝛿+min(1,𝛿)𝑓_𝐶𝐵 (3.23) whence we have the result.

Remark 3.4. Note that when 𝑛→∞, then 𝜆_𝑛, 𝜁_𝑛, and 𝛿_𝑛 tend to zero in Theorems 3.1–3.3 under the assumption (2.8).

4. Special Cases and Further Properties

Gould-Hopper polynomials g_𝑘^𝑑+1(𝑥,ℎ) [16], which are 𝑑-orthogonal polynomial sets of Hermite type [17], are generated by 𝑒^{ℎ𝑡𝑑+1}exp(𝑥𝑡)=^∞_𝑘=0g_𝑘^𝑑+1()𝑡𝑥,ℎ^𝑘𝑘! (4.1) from which it follows that g_𝑘^𝑑+1(𝑥,ℎ)=^{[𝑘/(𝑑+1)]}_𝑚=0𝑘!(()𝑚)!ℎ𝑚!𝑘−𝑑+1^𝑚𝑥^{𝑘−(𝑑+1)𝑚}, (4.2) where, as usual, [⋅] denotes the integer part.

In [4], the authors showed that the Gould-Hopper polynomials are Brenke-type polynomials with 𝐴(𝑡)=𝑒^{ℎ𝑡𝑑+1} and 𝐵(𝑡)=𝑒^𝑡, and the restrictions (1.9) and condition (2.8) for the operators given by (1.10) are satisfied under the assumption ℎ≥0. These operators including the Gould-Hopper polynomials are as follows: 𝐿^∗_𝑛𝑓;𝑥∶=𝑒^{∞−𝑛𝑥−ℎ}_𝑘=0g_𝑘^𝑑+1(𝑛𝑥,ℎ)𝑓𝑘𝑘!𝑛, (4.3) where 𝑥∈[0,∞).

The special case 𝐴(𝑡)=𝑒^{ℎ𝑡𝑑+1} and 𝐵(𝑡)=𝑒^𝑡 of (1.11) gives the following Kantorovich version of 𝐾_𝑛(𝑓;𝑥) including the Gould-Hopper polynomials: 𝐾^∗_𝑛𝑓;𝑥∶=𝑛𝑒^{∞−𝑛𝑥−ℎ}_𝑘=0g_𝑘^𝑑+1(𝑛𝑥,ℎ)𝑘!^{(𝑘+1)/𝑛}_𝑘/𝑛𝑓(𝑡)𝑑𝑡 (4.4) under the assumption ℎ≥0.

Remark 4.1. For ℎ=0, we find g_𝑘^𝑑+1(𝑛𝑥,0)=(𝑛𝑥)^𝑘 and the operators given by (4.4) reduce to the Szasz-Mirakyan-Kantorovich operators given by (1.12).

Now, we give a Voronovskaya-type theorem for the operators (4.4). In order to prove this theorem, we need the following lemmas.

Lemma 4.2. For the operators 𝐾^∗_𝑛, one has 𝐾^∗_𝑛()𝐾1;𝑥=1,_∗𝑛()ℎ()𝑠;𝑥=𝑥+𝑑+1𝑛+1,𝐾2𝑛_∗𝑛𝑠²;𝑥=𝑥²+2𝑛(ℎ())𝑥+1𝑑+1+1𝑛²ℎ(ℎ()}+1𝑑+1){𝑑+1+𝑑+23,𝐾^∗_𝑛𝑠³;𝑥=𝑥³+3𝑥²𝑛ℎ()+3𝑑+12+𝑥𝑛²3ℎ²(𝑑+1)²()+7+3ℎ𝑑+1)(𝑑+32+1𝑛³ℎ²(ℎ+3)(𝑑+1)³+ℎ(𝑑+1)³+32ℎ(ℎ+1)(𝑑+1)²()+1+ℎ𝑑+14,𝐾^∗_𝑛𝑠⁴;𝑥=𝑥⁴+4𝑥³𝑛{ℎ(𝑑+1)+2}+3𝑥²𝑛²2ℎ(ℎ+1)(𝑑+1)²()+𝑥+6ℎ𝑑+1+5𝑛³12ℎ²(𝑑+1)²(𝑑+2)+4ℎ³(𝑑+1)³+2ℎ(𝑑+1)2𝑑²+1+10𝑑+15+6𝑛⁴ℎ³(ℎ+6)(𝑑+1)⁴+8ℎ(1+3ℎ)(𝑑+1)³−9ℎ(ℎ+1)(𝑑+1)²+7ℎ(𝑑+1)+2ℎ³(𝑑+1)³+4𝑑(𝑑−1)(𝑑+1)²ℎ²+3ℎ²𝑑²(𝑑+1)²+()𝑑()1𝑑−2)(𝑑−1𝑑+1ℎ+5. (4.5)

Proof. From the generating function (4.1) for the Gould-Hopper polynomials, one can easily find the above equalities.

Lemma 4.3. For 𝑥∈[0,∞), one has 𝐾^∗_𝑛(𝑠−𝑥)²=𝑥;𝑥𝑛+1𝑛²ℎ(ℎ()}+1𝑑+1){𝑑+1+𝑑+23𝐾^∗_𝑛(𝑠−𝑥)⁴;𝑥=3𝑥²𝑛²+𝑥𝑛³6ℎ(ℎ+2)(𝑑+1)²()+1+2ℎ𝑑+1)(−3𝑑+2+5𝑛⁴ℎ³(ℎ+6)(𝑑+1)⁴+2ℎℎ²+12ℎ+4(𝑑+1)³+7ℎ(𝑑+1)−9ℎ(ℎ+1)(𝑑+1)²+4𝑑(𝑑−1)(𝑑+1)²ℎ²+3ℎ²𝑑²(𝑑+1)²+()𝑑()1𝑑−2)(𝑑−1𝑑+1ℎ+5. (4.6)

Proof. It is enough to use Lemma 4.2 to obtain above equalities.

Theorem 4.4. Let 𝑓∈𝐶²[0,𝑎]. Then one has lim_𝑛→∞𝑛𝐾^∗_𝑛𝑓;𝑥−𝑓(𝑥)=𝑓^′(𝑥ℎ()+1)𝑑+12+𝑥𝑓^′′(𝑥).2! (4.7)

Proof. By Taylor's theorem, we get 𝑓(𝑠)=𝑓(𝑥)+(𝑠−𝑥)𝑓^′(𝑥)+(𝑠−𝑥)²𝑓2!^′′(𝑥)+(𝑠−𝑥)²𝜂(𝑠;𝑥), (4.8) where 𝜂(𝑠;𝑥)∈𝐶[0,𝑎] and lim_𝑠→𝑥𝜂(𝑠;𝑥)=0. If we apply the operator 𝐾^∗_𝑛 to the both sides of (4.8), we obtain 𝐾^∗_𝑛𝑓;𝑥=𝑓(𝑥)+𝑓^′(𝑥)𝐾^∗_𝑛()+𝑓𝑠−𝑥;𝑥_′′(𝑥)𝐾2!^∗𝑛(𝑠−𝑥)²;𝑥+𝐾^∗_𝑛(𝑠−𝑥)²𝜂(𝑠;𝑥);𝑥. (4.9) In view of Lemmas 4.2 and 4.3, the equality (4.9) can be written in the form 𝑛𝐾^∗_𝑛(𝑥ℎ()𝑓;𝑥−𝑓)=𝑛𝑑+1𝑛+1𝑓2𝑛^′(𝑥)𝑥+𝑛𝑛+1𝑛²ℎ(ℎ()}+1𝑑+1){𝑑+1+𝑑+23𝑓^′′(𝑥)2!+𝑛𝐾^∗_𝑛(𝑠−𝑥)²𝜂(𝑠;𝑥);𝑥, (4.10) where 𝐾^∗_𝑛(𝑠−𝑥)²𝜂(𝑠;𝑥);𝑥=𝑛𝑒^{∞−𝑛𝑥−ℎ}_𝑘=0g_𝑘^𝑑+1(𝑛𝑥,ℎ)𝑘!^{(𝑘+1)/𝑛}_𝑘/𝑛(𝑠−𝑥)²𝜂(𝑠;𝑥)𝑑𝑠. (4.11) Applying Cauchy-Schwarz inequality, we get 𝑛𝐾^∗_𝑛(𝑠−𝑥)²𝜂(𝑠;𝑥);𝑥≤𝑛²𝑒^{∞−𝑛𝑥−ℎ}_𝑘=0g_𝑘^𝑑+1(𝑛𝑥,ℎ)𝑘!^{(𝑘+1)/𝑛}_𝑘/𝑛(𝑠−𝑥)⁴𝑑𝑠^{1/2(𝑘+1)/𝑛}_𝑘/𝑛𝜂²(𝑠;𝑥)𝑑𝑠^1/2. (4.12) If we use Cauchy-Schwarz inequality again on the right-hand side of the inequality above, then we conclude that 𝑛𝐾^∗_𝑛(𝑠−𝑥)²𝜂(𝑠;𝑥);𝑥≤𝑛³𝑒^{∞−𝑛𝑥−ℎ}_𝑘=0g_𝑘^𝑑+1(𝑛𝑥,ℎ)𝑘!^{(𝑘+1)/𝑛}_𝑘/𝑛(𝑠−𝑥)⁴𝑑𝑠^1/2⋅𝑛𝑒^{∞−𝑛𝑥−ℎ}_𝑘=0g_𝑘^𝑑+1(𝑛𝑥,ℎ)𝑘!^{(𝑘+1)/𝑛}_𝑘/𝑛𝜂²(𝑠;𝑥)𝑑𝑠^1/2=𝑛²𝐾^∗_𝑛(𝑠−𝑥)⁴;𝑥𝐾^∗_𝑛𝜂²(𝑠;𝑥);𝑥. (4.13) In view of Lemma 4.3, lim_𝑛→∞𝑛²𝐾^∗_𝑛(𝑠−𝑥)⁴;𝑥=3𝑥² (4.14) holds. On the other hand, since 𝜂(𝑠;𝑥)∈𝐶[0,𝑎] and lim_𝑠→𝑥𝜂(𝑠;𝑥)=0, then it follows from Theorem 2.3 that lim_𝑛→∞𝐾^∗_𝑛𝜂²(𝑠;𝑥);𝑥=𝜂²(𝑥;𝑥)=0. (4.15) Considering (4.13), (4.14), and (4.15), we immediately see that lim_𝑛→∞𝑛𝐾^∗_𝑛(𝑠−𝑥)²𝜂(𝑠;𝑥);𝑥=0. (4.16) Then, taking limit as 𝑛→∞ in (4.10) and using (4.16), we have lim_𝑛→∞𝑛𝐾^∗_𝑛𝑓;𝑥−𝑓(𝑥)=𝑓^′(𝑥ℎ()+1)𝑑+12+𝑥𝑓^′′(𝑥)2! (4.17) which completes the proof.

Remark 4.5. Getting ℎ=0 in Theorem 4.4 gives a Voronovskaya-type result for the Szasz-Mirakyan-Kantorovich operators given by (1.12).