A Kantorovich-Stancu Type Generalization of Szasz Operators including Brenke Type Polynomials

Abstract

We introduce a Kantorovich-Stancu type modification of a generalization of Szasz operators defined by means of the Brenke type polynomials and obtain approximation properties of these operators. Also, we give a Voronovskaya type theorem for Kantorovich-Stancu type operators including Gould-Hopper polynomials.

1. Introduction

For each positive 𝑛 and 𝑓∈𝐶_𝐵([0,∞)) or 𝐶([0,∞))∩𝐸, the Szasz-Mirakyan operators defined by 𝑆_𝑛𝑓;𝑥∶=𝑒^{∞−𝑛𝑥}_𝑘=0(𝑛𝑥)^𝑘𝑓𝑘𝑘!𝑛 (1) have an important role in the approximation theory [1]. Their Korovkin type approximation properties and rates of convergence have been investigated by many researchers. Recently, there is a growing interest in defining linear positive operators via special functions (see [2–13]). In particular, many authors have studied various generalizations of Szasz operators via special functions. In [14], Jakimovski and Leviatan constructed a generalization of Szasz operators by means of the Appell polynomials. Then, Ismail [15] presented another generalization of Szasz operators by means of Sheffer polynomials, which involves the operators (1) defined by Jakimovski and Leviatan in [14]. In [11],Varma et al. considered the following generalization of Szasz operators by means of the Brenke type polynomials, which are motivated by the operators defined by Jakimovski and Leviatanand Ismail, for 𝑥≥0 and 𝑛∈ℕ: 𝐿_𝑛1𝑓;𝑥∶=𝐴(1)𝐵(𝑛𝑥)^∞_𝑘=0𝑝_𝑘()𝑓𝑘𝑛𝑥𝑛 (2) under the following assumptions: (i)𝐴(1)𝑎≠0,^𝑘−𝑟𝑏_𝑟𝐴(1)()[)⟶(),(4)(5)|𝑡|(),≥0,0≤𝑟≤𝑘,𝑘=0,1,2,…,ii𝐵∶0,∞0,∞iii)(andconvergefor<𝑅𝑅>1 (3) where 𝐴(𝑡)=^∞_𝑟=0𝑎_𝑟𝑡^𝑟,𝑎₀𝐵(𝑡)=≠0,_∞𝑟=0𝑏_𝑟𝑡^𝑟,𝑏_𝑟≠0(𝑟≥0) (4) are analytic functions and the Brenke type polynomials [16] have generating functions of the form 𝐴(𝑡)𝐵(𝑥𝑡)=^∞_𝑘=0𝑝_𝑘(𝑥)𝑡^𝑘, (5) where 𝑝_𝑘(𝑥)=^𝑘_𝑟=0𝑎_𝑘−𝑟𝑏_𝑟𝑥^𝑟,𝑘=0,1,2,…. (6)

The Kantorovich type of Szasz-Mirakyan operators is defined by [17] 𝐾_𝑛𝑓;𝑥∶=𝑛𝑒^{∞−𝑛𝑥}_𝑘=0(𝑛𝑥)^𝑘𝑘!^{(𝑘+1)/𝑛}_𝑘/𝑛𝑓(𝑡)𝑑𝑡. (7) The approximation properties of the Szasz-Mirakyan-Kantorovich operators and their various iterates were studied by many authors in [12, 18–23].

Recently, in [8], the Kantorovich type of the operators given by (2) under the assumptions (3) has been defined as 𝐾_𝑛𝑛𝑓;𝑥∶=𝐴(1)𝐵(𝑛𝑥)^∞_𝑘=0𝑝_𝑘(𝑛𝑥)^{(𝑘+1)/𝑛}_𝑘/𝑛𝑓(𝑡)𝑑𝑡, (8) where 𝑛∈ℕ, 𝑥≥0 and 𝑓∈𝐶[0,∞), and some of its properties have been investigated.

The purpose of this study is to introduce a Kantorovich-Stancu type modification of the operators given by (8) and to examine the approximation properties of these operators. We also present a Kantorovich-Stancu type of the operators including Gould-Hopper polynomials and then we prove a Voronovskaya type theorem for these operators including Gould-Hopper polynomials.

2. Construction of the Operators

For each positive integer 𝑛, 𝑥≥0 and 𝑓∈𝐶_𝐵([0,∞)), or 𝐶([0,∞))∩𝐸, let us consider the following operators: 𝐾_𝑛^(𝛼,𝛽)𝑓;𝑥∶=𝑛+𝛽𝐴(1)𝐵(𝑛𝑥)^∞_𝑘=0𝑝_𝑘(𝑛𝑥)₍₎^{(𝑘+𝛼+1)/(𝑛+𝛽)}_{𝑘+𝛼/(𝑛+𝛽)}𝑓(𝑡)𝑑𝑡, (9) where 𝛼 and 𝛽 parameters satisfy the condition 0≤𝛼≤𝛽. For the approximation properties of Stancu type operators, we refer to [24–27].

It is clear that for 𝛼=𝛽=0,𝐾_𝑛^(𝛼,𝛽)(𝑓;𝑥) reduces to the operators defined by (8).

In the case of 𝐵(𝑡)=𝑒^𝑡 and 𝐴(𝑡)=1,with the help of (5) it follows that 𝑝_𝑘(𝑥)=𝑥^𝑘/𝑘!. So the operator 𝐾_𝑛^(𝛼,𝛽)(𝑓;𝑥) gives the Kantorovich-Stancu type of Szasz-Mirakyan operators as follows: 𝐾_𝑛^(𝛼,𝛽)𝑓;𝑥∶=𝑛+𝛽𝑒^{∞−𝑛𝑥}_𝑘=0(𝑛𝑥)^𝑘𝑘!₍₎^{(𝑘+𝛼+1)/(𝑛+𝛽)}_{𝑘+𝛼/(𝑛+𝛽)}𝑓(𝑡)𝑑𝑡, (10) where 𝛼 and 𝛽 parameters satisfy the condition 0≤𝛼≤𝛽.

In the case of 𝛼=𝛽=0, the operator (10) turns out to be the Szasz-Mirakyan-Kantorovich operators given by (7).

For 𝐵(𝑡)=𝑒^𝑡,𝐾_𝑛^(𝛼,𝛽)(𝑓;𝑥) gives the Kantorovich-Stancu type of the operators 𝑃_𝑛(𝑓;𝑥) proposed by Jakimovski and Leviatan in [14].

Now, for the operators 𝐾_𝑛^(𝛼,𝛽) given by (9), we give some results which are necessary to prove the main theorem.

Lemma 1. Kantorovich-Stancu type operators, defined by (9), are linear and positive.

Lemma 2. For each 𝑥∈[0,∞), the Kantorovich-Stancu type operators (9) have the following properties: 𝐾_𝑛^(𝛼,𝛽)()𝐾1;𝑥=1,_{𝑛(𝛼,𝛽)}()=𝑛𝑠;𝑥𝐵𝑛+𝛽^′(𝑛𝑥)𝐵()𝐴𝑛𝑥𝑥+^′(1)𝐴(1)+𝑛+𝛽2𝛼+12,𝐾𝑛+𝛽_{𝑛(𝛼,𝛽)}𝑠²=𝑛;𝑥𝑛+𝛽²𝐵^′′(𝑛𝑥)𝐵()𝑥𝑛𝑥²+𝑛𝐵^′(𝑛𝑥)2𝐴^′(1)+(2𝛼+2)𝐴(1)𝑛+𝛽²𝐴(1)𝐵()𝑥+1𝑛𝑥𝑛+𝛽²𝐴(1)𝐴^′′(1)+(2𝛼+2)𝐴^′(1)+𝛼²1+𝛼+3𝐴(1). (11)(12)(13)

Proof. From the generating function of the Brenke type polynomials given by (5), a few calculations reveal that ^∞_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)(1(𝐴+𝐴)+𝐵𝑛𝑥)^′′(1)+𝐴^′(1). (14) By using these equalities, we obtain the assertions of the lemma by simple calculation.

Lemma 3. For each 𝑥∈[0,∞), one has 𝐾_𝑛^(𝛼,𝛽)(𝑠−𝑥)²=𝑛;𝑥𝑛+𝛽²𝐵^′′(𝑛𝑥)𝐵()−𝑛𝑥2𝑛𝐵^′(𝑛𝑥)𝐵()𝑥𝑛+𝛽𝑛𝑥+1²+𝑛𝐵^′(𝑛𝑥)2𝐴^′(1)+𝐴(1)𝑛+𝛽²𝐴(1)𝐵()+()𝑛𝑥2𝛼+1𝑛𝐵^′(𝑛𝑥)𝑛+𝛽²𝐵()−𝑛𝑥2𝐴^′(1)𝐴(1)−𝑛+𝛽2𝛼+1𝐴𝑛+𝛽𝑥+^′′(1)+𝐴^′(1)𝑛+𝛽²𝐴(1)+(2𝛼+1)𝐴^′(1)𝑛+𝛽²𝐴(1)+𝛼²+𝛼+(1/3)𝑛+𝛽². (15)

Theorem 4. Let [),𝑓(𝑥)𝐸∶=𝑓∶𝑥∈0,∞1+𝑥²𝑖𝑠𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑎𝑠𝑥⟶∞,lim_𝑦→∞𝐵^′𝑦𝐵𝑦=1,lim_𝑦→∞𝐵^′′𝑦𝐵𝑦=1. (16) If 𝑓∈𝐶[0,∞)∩𝐸, then lim_𝑛→∞𝐾_𝑛^(𝛼,𝛽)𝑓;𝑥=𝑓(𝑥), (17) and the operators 𝐾_𝑛^(𝛼,𝛽) converge uniformly in each compact subset of [0,∞).

Proof. According to Lemma 2, by considering the equality (16), we get lim_𝑛→∞𝐾_𝑛^(𝛼,𝛽)𝑠^𝑖;𝑥=𝑥^𝑖,𝑖=0,1,2. (18) This convergence is satisfied uniformly in each compact subset of [0,∞). Then, the proof follows from the universal Korovkin-type property (vi) of Theorem 4.1.4 in [28].

3. Rates of Convergence

In this section, we compute the rates of convergence of the operators 𝐾_𝑛^(𝛼,𝛽)(𝑓) to 𝑓 by means of a classical approach, the second modulus of continuity, and Peetre’s 𝐾-functional.

Let 𝑓∈𝐶[0,∞). Then for 𝛿>0, the modulus of continuity of 𝑓 denoted by 𝑤(𝑓;𝛿) is defined to be 𝑤𝑓;𝛿∶=sup_{𝑥,𝑦∈[0,∞)𝑥−𝑦≤𝛿}𝑓(𝑥)−𝑓𝑦, (19) where 𝐶[0,∞)denotes the space of uniformly continuous functions on [0,∞).Then, for any 𝛿>0 and each 𝑥∈[0,∞), it is well known that one can write 𝑓(𝑥)−𝑓𝑦≤𝑤𝑓;𝛿𝑥−𝑦𝛿+1. (20)

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

Theorem 5. For 𝑓∈𝐶[0,∞)∩𝐸, one has 𝐾_𝑛^(𝛼,𝛽)𝑓;𝑥−𝑓(𝑥)≤2𝑤𝑓;𝜆_𝑛(𝑥), (21) where 𝜆=𝜆_𝑛(𝑥)=𝐾_𝑛^(𝛼,𝛽)(𝑠−𝑥)²=𝑛;𝑥𝑛+𝛽²𝐵^′′(𝑛𝑥)𝐵()−𝑛𝑥2𝑛𝐵^′(𝑛𝑥)𝐵()𝑥𝑛+𝛽𝑛𝑥+1²+𝑛𝐵^′(𝑛𝑥)2𝐴^′(1)+𝐴(1)𝑛+𝛽²𝐴(1)𝐵()+()𝑛𝑥2𝛼+1𝑛𝐵^′(𝑛𝑥)𝑛+𝛽²𝐵()−𝑛𝑥2𝐴^′(1)𝐴(1)−𝑛+𝛽2𝛼+1𝐴𝑛+𝛽𝑥+^′′(1)+𝐴^′(1)𝑛+𝛽²𝐴(1)+(2𝛼+1)𝐴^′(1)𝑛+𝛽²𝐴(1)+𝛼²+𝛼+(1/3)𝑛+𝛽². (22)

Proof. Using linearity of the operators 𝐾_𝑛^(𝛼,𝛽), (11) and (20), we get 𝐾_𝑛^(𝛼,𝛽)(𝑥≤𝑓;𝑥−𝑓)𝑛+𝛽𝐴(1)𝐵(𝑛𝑥)^∞_𝑘=0𝑃_𝑘()×𝑛𝑥_{()(𝑘+𝛼+1)/(𝑛+𝛽)𝑘+𝛼/(𝑛+𝛽)}𝑓(𝑠)(𝑥≤−𝑓)𝑑𝑠𝑛+𝛽𝐴(1)𝐵(𝑛𝑥)^∞_𝑘=0𝑃_𝑘(𝑛𝑥)×₍₎^{(𝑘+𝛼+1)/(𝑛+𝛽)}_{𝑘+𝛼/(𝑛+𝛽)}|𝑠−𝑥|𝛿𝑤≤+1𝑓;𝛿𝑑𝑠1+𝑛+𝛽𝐴(1)𝐵(𝑛𝑥)𝛿^∞_𝑘=0𝑃_𝑘(𝑛𝑥)×₍₎^{(𝑘+𝛼+1)/(𝑛+𝛽)}_{𝑘+𝛼/(𝑛+𝛽)}|𝑠−𝑥|𝑑𝑠𝑤𝑓;𝛿. (23) According to the Cauchy-Schwarz inequality for integration, we obtain that ₍₎^{(𝑘+𝛼+1)/(𝑛+𝛽)}_{𝑘+𝛼/(𝑛+𝛽)}||≤1𝑠−𝑥𝑑𝑠𝑛+𝛽^{()(𝑘+𝛼+1)/(𝑛+𝛽)𝑘+𝛼/(𝑛+𝛽)}|𝑠−𝑥|²𝑑𝑠^1/2 (24) from which, it follows that ^∞_𝑘=0𝑃_𝑘(𝑛𝑥)₍₎^{(𝑘+𝛼+1)/(𝑛+𝛽)}_{𝑘+𝛼/(𝑛+𝛽)}||≤1𝑠−𝑥𝑑𝑠𝑛+𝛽^∞𝑘=0𝑃_𝑘(𝑛𝑥)₍₎^{(𝑘+𝛼+1)/(𝑛+𝛽)}_{𝑘+𝛼/(𝑛+𝛽)}|𝑠−𝑥|²𝑑𝑠^1/2. (25) By using the Cauchy-Schwarz inequality for summation on the right hand side of (25), we may write ^∞_𝑘=0𝑃_𝑘(𝑛𝑥)₍₎^{(𝑘+𝛼+1)/(𝑛+𝛽)}_{𝑘+𝛼/(𝑛+𝛽)}||≤𝑠−𝑥𝑑𝑠𝐴(1)𝐵(𝑛𝑥)𝐴(1)𝐵()𝑛+𝛽𝑛𝑥𝐾𝑛+𝛽^{𝑛(𝛼,𝛽)}(𝑠−𝑥)²;𝑥^1/2=𝐴(1)𝐵(𝑛𝑥)𝐾𝑛+𝛽^{𝑛(𝛼,𝛽)}(𝑠−𝑥)²;𝑥^1/2=𝐴(1)𝐵(𝑛𝑥)𝜆𝑛+𝛽^𝑛(𝑥)^1/2, (26) where 𝜆_𝑛(𝑥) is given by (22). Considering this inequality in (23), we find that 𝐾_𝑛^(𝛼,𝛽)(𝑥≤1𝑓;𝑥−𝑓)1+𝛿𝜆_𝑛(𝑥)𝑤𝑓;𝛿. (27) If we set 𝛿=𝜆_𝑛(𝑥), the proof is completed.

Now, we will study the rates of convergence of the operators 𝐾_𝑛^(𝛼,𝛽) to 𝑓 by means of the second modulus of continuity and Peetre’s 𝐾-functional.

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, (28) 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, (29) where 𝐶²_𝐵[0,∞)∶=𝑔∈𝐶_𝐵[0,∞)∶𝑔^′,𝑔^′′∈𝐶_𝐵[0,∞) (30) and the norm ‖𝑔‖_𝐶2𝐵∶=‖𝑔‖_𝐶𝐵+‖𝑔^′‖_𝐶𝐵+‖𝑔^′′‖_𝐶𝐵 (see [29]). It is clear that the following inequality: 𝐾𝑓;𝛿≤𝑀𝑤₂𝑓;√𝛿+min(1,𝛿)𝑓_𝐶𝐵 (31) holds for all 𝛿>0. The constant 𝑀 is independent of 𝑓 and 𝛿.

Theorem 6. Let 𝑓∈𝐶²_𝐵[0,∞). If 𝐾_𝑛^(𝛼,𝛽) is defined by (9), then one has 𝐾_𝑛^(𝛼,𝛽)𝑓;𝑥−𝑓(𝑥)≤𝜁𝑓_𝐶2𝐵, (32) where 𝜁=𝜁_𝑛(𝑥)=𝑛𝑛+𝛽²𝐵^′′(𝑛𝑥)()−2𝐵𝑛𝑥𝑛𝐵^′(𝑛𝑥)𝐵()+1𝑛+𝛽𝑛𝑥2𝑥²+𝑛𝐵^′(𝑛𝑥)2𝐴^′(1)+(2𝛼+2)𝐴(1)2𝑛+𝛽²𝐴(1)𝐵()−𝑛𝑥2𝐴^′(1)+(2𝛼+1)𝐴(1)2𝐴(1)+𝑛𝑛+𝛽𝐵𝑛+𝛽^′(𝑛𝑥)𝐵()𝑥+𝐴𝑛𝑥−1_′′(1)+𝐴^′(1)2𝑛+𝛽²𝐴(1)+(2𝛼+1)𝐴^′(1)2𝑛+𝛽²𝐴(1)+𝛼²+𝛼+(1/3)2𝑛+𝛽²+2𝐴^′(1)+(2𝛼+1)𝐴(1)2𝐴(1).𝑛+𝛽 (33)

Proof. We can write from the Taylor expansion of 𝑓, the linearity of the operators 𝐾_𝑛^(𝛼,𝛽), and (11) 𝐾_𝑛^(𝛼,𝛽)𝑓;𝑥−𝑓(𝑥)=𝑓^′(𝑥)𝐾_𝑛^(𝛼,𝛽)()+1𝑠−𝑥;𝑥2𝑓^′′𝜂𝐾_𝑛^(𝛼,𝛽)(𝑠−𝑥)²;𝑥,𝜂∈(𝑥,𝑠). (34) From Lemma 2, it is obvious that 𝐾_𝑛^(𝛼,𝛽)()=𝑛𝑠−𝑥;𝑥𝐵𝑛+𝛽^′(𝑛𝑥)𝐵()𝐴𝑛𝑥−1𝑥+^′(1)𝐴(1)+𝑛+𝛽2𝛼+12𝑛+𝛽≥0 (35) for 𝑠≥𝑥. Thus, by considering Lemmas 2 and 3 in (34), one can write 𝐾_𝑛^(𝛼,𝛽)(𝑥≤𝑛𝑓;𝑥−𝑓)𝐵𝑛+𝛽^′(𝑛𝑥)𝐵()𝑥+𝐴𝑛𝑥−1_′(1)𝐴(1)+𝑛+𝛽2𝛼+12𝑓𝑛+𝛽^′𝐶_𝐵+12𝑛𝑛+𝛽²𝐵^′′(𝑛𝑥)𝐵()−𝑛𝑥2𝑛𝐵^′(𝑛𝑥)𝐵()𝑥𝑛+𝛽𝑛𝑥+1²+𝑛𝐵^′(𝑛𝑥)2𝐴^′(1)+𝐴(1)𝑛+𝛽²𝐴(1)𝐵()+()𝑛𝑥2𝛼+1𝑛𝐵^′(𝑛𝑥)𝑛+𝛽²𝐵()−𝑛𝑥2𝐴^′(1)𝐴(1)−𝑛+𝛽2𝛼+1𝑥+𝐴𝑛+𝛽_′′(1)+𝐴^′(1)𝑛+𝛽²𝐴(1)+(2𝛼+1)𝐴^′(1)𝑛+𝛽²𝐴(1)+𝛼²+𝛼+(1/3)𝑛+𝛽²𝑓^′′_𝐶𝐵≤𝑛𝑛+𝛽²𝐵^′′(𝑛𝑥)()−2𝐵𝑛𝑥𝑛𝐵^′(𝑛𝑥)𝐵()+1𝑛+𝛽𝑛𝑥2𝑥²+𝑛𝐵^′(𝑛𝑥)2𝐴^′(1)+(2𝛼+2)𝐴(1)2𝑛+𝛽²𝐴(1)𝐵()−𝑛𝑥2𝐴^′(1)+(2𝛼+1)𝐴(1)2𝐴(1)+𝑛𝑛+𝛽𝐵𝑛+𝛽^′(𝑛𝑥)𝐵()𝑥+𝐴𝑛𝑥−1_′′(1)+𝐴^′(1)2𝑛+𝛽²𝐴(1)+(2𝛼+1)𝐴^′(1)2𝑛+𝛽²𝐴(1)+𝛼²+𝛼+(1/3)2𝑛+𝛽²+2𝐴^′(1)+(2𝛼+1)𝐴(1)2𝐴(1)𝑓𝑛+𝛽^𝐶2𝐵 (36) which completes the proof.

Theorem 7. If 𝑓∈𝐶_𝐵[0,∞), then one has 𝐾_𝑛^(𝛼,𝛽)(𝑥𝑤𝑓;𝑥−𝑓)≤2𝑀₂𝑓;√𝛿+min(1,𝛿)𝑓_𝐶𝐵, (37) where 𝛿∶=𝛿_𝑛(𝑥)=12𝜁_𝑛(𝑥) (38) and 𝑀>0 is a constant which is independent of the function 𝑓 and 𝛿. Also, 𝜁_𝑛(𝑥) is the same as in Theorem 6.

Proof. Suppose that 𝑔∈𝐶²_𝐵[0,∞). From Theorem 6, we have 𝐾_𝑛^(𝛼,𝛽)(𝑥≤𝐾𝑓;𝑥−𝑓)_{𝑛(𝛼,𝛽)}𝑓−𝑔;𝑥+𝐾_𝑛^(𝛼,𝛽)(𝑥+𝑔(𝑥)(𝑥𝑔;𝑥−𝑔)−𝑓)≤2𝑓−𝑔_𝐶𝐵+𝜁𝑔_𝐶2𝐵=2𝑓−𝑔_𝐶𝐵+𝛿𝑔_𝐶2𝐵. (39) Since the left-hand side of inequality (39) does not depend on the function 𝑔∈𝐶²_𝐵[0,∞), we get 𝐾_𝑛^(𝛼,𝛽)𝑓;𝑥−𝑓(𝑥)≤2𝐾𝑓;𝛿, (40) where 𝐾(𝑓;𝛿) is Peetre’s 𝐾-functional defined by (29). By using the relation (31) in (39), the inequality 𝐾_𝑛^(𝛼,𝛽)𝑓;𝑥−𝑓(𝑥)≤2𝑀𝑤₂𝑓;√𝛿+min(1,𝛿)𝑓_𝐶𝐵 (41) holds.

Remark 8. In Theorems 5–7, 𝜆_𝑛,𝜁_𝑛,𝛿_𝑛→0 when 𝑛→∞ under the assumption (16).

4. Special Cases of the Operators 𝐾_𝑛^(𝛼,𝛽) and Further Properties

Gould-Hopper polynomials 𝑔_𝑘^𝑑+1(𝑥,ℎ)are defined through the identity 𝑔_𝑘^𝑑+1(𝑥,ℎ)=^{[𝑘/(𝑑+1)]}_𝑚=0𝑘!(()𝑚)!ℎ𝑚!𝑘−𝑑+1^𝑚𝑥^{𝑘−(𝑑+1)𝑚} (42) and satisfy the generating function 𝑒^{ℎ𝑡𝑑+1}exp(𝑥𝑡)=^∞_𝑘=0𝑔_𝑘^𝑑+1()𝑡𝑥,ℎ^𝑘,𝑘! (43) where, as usual, [⋅] denotes the integer part [30].

The Gould-Hopper polynomials are Brenke-type polynomials for the special case of 𝐴(𝑡)=𝑒^{ℎ𝑡𝑑+1} and 𝐵(𝑡)=𝑒^𝑡 in (5). From (2), the operators including the Gould-Hopper polynomials are as follows: 𝐿^∗_𝑛𝑓;𝑥∶=𝑒^{∞−𝑛𝑥−ℎ}_𝑘=0𝑔_𝑘^𝑑+1(𝑛𝑥,ℎ)𝑓𝑘𝑘!𝑛, (44) where 𝑥∈[0,∞) and ℎ≥0 (see [11]).

Similarly, the special case 𝐴(𝑡)=𝑒^{ℎ𝑡𝑑+1} and 𝐵(𝑡)=𝑒^𝑡 of (9) gives the following Kantorovich-Stancu type operators 𝐾_𝑛^{∗(𝛼,𝛽)}(𝑓;𝑥) including the Gould-Hopper polynomials: 𝐾_𝑛^{∗(𝛼,𝛽)}𝑓;𝑥∶=𝑛+𝛽𝑒^{∞−𝑛𝑥−ℎ}_𝑘=0𝑔_𝑘^𝑑+1(𝑛𝑥,ℎ)×𝑘!_{()(𝑘+𝛼+1)/(𝑛+𝛽)𝑘+𝛼/(𝑛+𝛽)}𝑓(𝑡)𝑑𝑡 (45) under the assumption ℎ≥0.

Remark 9. For ℎ=0, we have 𝑔_𝑘^𝑑+1(𝑛𝑥,0)=(𝑛𝑥)^𝑘 and the operators given by (45) reduce to the Kantorovich-Stancu type of Szasz-Mirakyan operators given by (10).

Remark 10. For 𝛼=𝛽=0, the operators (45) give the Kantorovich type operators including the Gould-Hopper polynomials given by 𝐾^∗_𝑛𝑓;𝑥∶=𝑛𝑒^{∞−𝑛𝑥−ℎ}_𝑘=0𝑔_𝑘^𝑑+1(𝑛𝑥,ℎ)𝑘!^{(𝑘+1)/𝑛}_𝑘/𝑛𝑓(𝑡)𝑑𝑡 (46) in [8].

Remark 11. For ℎ=0 in Remark 10, we get 𝑔_𝑘^𝑑+1(𝑛𝑥,0)=(𝑛𝑥)^𝑘 and then the operators given by (46) reduce to the Szasz-Mirakyan-Kantorovich operators given by (7).

Now, in order to prove a Voronovskaya type theorem for the operators given by (45), let us prove the following lemmas.

Lemma 12. For the operators 𝐾_𝑛^{∗(𝛼,𝛽)}, one has 𝐾_𝑛^{∗(𝛼,𝛽)}()𝐾1;𝑥=1,_{𝑛∗(𝛼,𝛽)}(𝑠;𝑥)=𝑛𝑥+ℎ()𝑛+𝛽𝑑+1+𝑛+𝛽2𝛼+12,𝐾𝑛+𝛽_{𝑛∗(𝛼,𝛽)}𝑠²=𝑛;𝑥₂𝑥²𝑛+𝛽²+𝑛𝑥𝑛+𝛽²×{()+(+12ℎ𝑑+12𝛼+2)}𝑛+𝛽²×ℎ(ℎ+1)(𝑑+1)²+()ℎ()+𝛼2𝛼+1𝑑+1₂1+𝛼+3,𝐾_𝑛^{∗(𝛼,𝛽)}𝑠³=𝑛;𝑥₃𝑥³𝑛+𝛽³+3𝑛²𝑥²2𝑛+𝛽³×{2𝛼+3+2(𝑑+1)ℎ}+𝑛𝑥2𝑛+𝛽³×6ℎ²(𝑑+1)²()×()+6ℎ𝑑+13+𝑑+2𝛼+12𝛼+6𝛼²+1+74𝑛+𝛽³4ℎ³(𝑑+1)³+6ℎ²(𝑑+1)²×(2𝛼+2𝑑+3)+4𝛼³+6𝛼²()×+4𝛼+1+2ℎ𝑑+12𝑑²+𝑑(2𝛼+7)+12𝛼+6𝛼²,𝐾+7_{𝑛∗(𝛼,𝛽)}𝑠⁴=𝑛;𝑥₄𝑥⁴𝑛+𝛽⁴+4𝑛³𝑥³𝑛+𝛽⁴(ℎ())+𝑑+1+𝛼+23𝑛²𝑥²𝑛+𝛽⁴2ℎ²(𝑑+1)²()×()+2ℎ𝑑+12𝛼+𝑑+4+2𝛼²++6𝛼+52𝑛𝑥𝑛+𝛽⁴2ℎ³(𝑑+1)³+6ℎ²(𝑑+1)²()()×𝛼+𝑑+2+ℎ𝑑+12𝑑²+10𝑑+6(3+𝑑)𝛼+6𝛼²+((+1+151+𝛼)(3+2𝛼2+𝛼))5𝑛+𝛽⁴×5ℎ(𝑑+1)⁴[((()1+ℎ7+ℎ6+ℎ))]+10ℎ𝑑+1³[()()1+ℎ3+ℎ)](1+2𝛼+10ℎℎ+1)(𝑑+1²((([(1+3𝛼1+𝛼))+5ℎ𝑑+1)(1+2𝛼2+𝛼3+2𝛼)])+5𝛼⁴+10𝛼³+10𝛼²+5𝛼+1. (47)

Proof. The proof follows from the generating function (43) for the Gould-Hopper polynomials.

Lemma 13. For each 𝑥∈[0,∞), one has 𝐾_𝑛^{∗(𝛼,𝛽)}(𝑠−𝑥)²=𝛽;𝑥₂𝑥²𝑛+𝛽²𝑛+𝑥𝑛+𝛽²{()+(−()2ℎ𝑑+12𝛼+2)}2ℎ𝑑+1+2𝛼+1+1𝑛+𝛽𝑛+𝛽²×ℎ(ℎ+1)(𝑑+1)²+(2𝛼+1)ℎ(𝑑+1)+𝛼²1+𝛼+3,𝐾_𝑛^{∗(𝛼,𝛽)}(𝑠−𝑥)⁴=𝛽;𝑥⁴𝑥⁴𝑛+𝛽⁴−𝑥³2𝛽²−3𝑛+(2ℎ(𝑑+1)+2𝛼+1)𝛽𝑛+𝛽⁴+𝑥²1𝑛+𝛽⁴3𝑛²−2𝑛𝛽(6ℎ(𝑑+1)+6𝛼+5)+23ℎ²(𝑑+1)²()×()()𝛽+3ℎ𝑑+12𝛼+𝑑+2+3𝛼1+𝛼+1²+𝑥2𝑛𝑛+𝛽⁴2ℎ³(𝑑+1)³+6ℎ²(𝑑+1)²×()()×𝛼+𝑑+2+ℎ𝑑+12𝑑²+10𝑑+6(3+𝑑)𝛼+6𝛼²+()×((−1+151+𝛼3+2𝛼2+𝛼))𝑛+𝛽³4ℎ³(𝑑+1)³+6ℎ²(𝑑+1)²×(2𝛼+2𝑑+3)+4𝛼³+6𝛼²()×+4𝛼+1+2ℎ𝑑+12𝑑²+𝑑(2𝛼+7)+6𝛼²+1+12𝛼+75𝑛+𝛽⁴5ℎ(𝑑+1)⁴[((()1+ℎ7+ℎ6+ℎ))]+10ℎ𝑑+1³[()()1+ℎ3+ℎ)](2𝛼+1+10𝑑+1²ℎ((([(1+ℎ)(1+3𝛼1+𝛼))+5ℎ𝑑+1)(1+2𝛼2+𝛼3+2𝛼)])+5𝛼⁴+10𝛼³+10𝛼²+5𝛼+1. (48)

Proof. From Lemma 12, the proof is obvious.

Theorem 14. Let 𝑓∈𝐶²[0,𝑎]. Then one has lim_𝑛→∞𝑛+𝛽𝐾_𝑛^{∗(𝛼,𝛽)}𝑓;𝑥−𝑓(𝑥)=𝑓^′(𝑥)𝛽𝑥+ℎ(𝑑+1)+2𝛼+12+𝑥𝑓^′′(𝑥).2! (49)

Proof. By Taylor’s theorem for 𝑓, we have 𝑓(𝑠)=𝑓(𝑥)+(𝑠−𝑥)𝑓^′(𝑥)+(𝑠−𝑥)²𝑓2!^′′(𝑥)+(𝑠−𝑥)²𝜂(𝑠;𝑥), (50) where 𝜂(𝑠;𝑥)∈𝐶[0,𝑎] and lim_𝑠→𝑥𝜂(𝑠;𝑥)=0. By applying the operator 𝐾_𝑛^{∗(𝛼,𝛽)} to the both sides of (50), we have 𝐾_𝑛^{∗(𝛼,𝛽)}𝑓;𝑥=𝑓(𝑥)+𝑓^′(𝑥)𝐾_𝑛^{∗(𝛼,𝛽)}()+𝑓𝑠−𝑥;𝑥_′′(𝑥)𝐾2!^{𝑛∗(𝛼,𝛽)}(𝑠−𝑥)²;𝑥+𝐾_𝑛^{∗(𝛼,𝛽)}(𝑠−𝑥)²𝜂(𝑠;𝑥);𝑥. (51) According to Lemmas 12 and 13, the equality (51) can be written as follows: 𝑛+𝛽𝐾_𝑛^{∗(𝛼,𝛽)}(𝑥=𝛽𝑓;𝑥−𝑓)𝑛+𝛽ℎ()𝑛+𝛽𝑥+𝑑+1+𝑛+𝛽2𝛼+12𝑓𝑛+𝛽^′(𝑥)+𝑛+𝛽𝑥²𝛽𝑛+𝛽²𝑛+𝑥𝑛+𝛽²{()+(−()2ℎ𝑑+12𝛼+2)}2ℎ𝑑+1+2𝛼+1+1𝑛+𝛽𝑛+𝛽²ℎ(ℎ+1)(𝑑+1)²+()()+𝛼2𝛼+1×ℎ𝑑+1²1+𝛼+3×𝑓^′′(𝑥)+𝐾2!𝑛+𝛽_𝑛^{∗(𝛼,𝛽)}(𝑠−𝑥)²𝜂(𝑠;𝑥);𝑥, (52) where 𝐾_𝑛^{∗(𝛼,𝛽)}(𝑠−𝑥)²𝜂()=𝑒𝑠;𝑥;𝑥𝑛+𝛽^{∞−𝑛𝑥−ℎ}_𝑘=0𝑔_𝑘^𝑑+1(𝑛𝑥,ℎ)×𝑘!_{()(𝑘+𝛼+1)/(𝑛+𝛽)𝑘+𝛼/(𝑛+𝛽)}(𝑠−𝑥)²𝜂(𝑠;𝑥)𝑑𝑠. (53) By applying Cauchy-Schwarz inequality, we can write 𝑛+𝛽𝐾_𝑛^{∗(𝛼,𝛽)}(𝑠−𝑥)²𝜂()≤𝑠;𝑥;𝑥𝑛+𝛽²𝑒^{∞−𝑛𝑥−ℎ}_𝑘=0𝑔_𝑘^𝑑+1(𝑛𝑥,ℎ)×𝑘!_{()(𝑘+𝛼+1)/(𝑛+𝛽)𝑘+𝛼/(𝑛+𝛽)}(𝑠−𝑥)⁴𝑑𝑠^1/2×₍₎^{(𝑘+𝛼+1)/(𝑛+𝛽)}_{𝑘+𝛼/(𝑛+𝛽)}𝜂²(𝑠;𝑥)𝑑𝑠^1/2. (54) If we consider Cauchy-Schwarz inequality again on the right-hand side of inequality above, then we arrive at 𝑛+𝛽𝐾_𝑛^{∗(𝛼,𝛽)}(𝑠−𝑥)²𝜂()≤𝑠;𝑥;𝑥𝑛+𝛽³𝑒^{∞−𝑛𝑥−ℎ}_𝑘=0𝑔_𝑘^𝑑+1(𝑛𝑥,ℎ)×𝑘!_{()(𝑘+𝛼+1)/(𝑛+𝛽)𝑘+𝛼/(𝑛+𝛽)}(𝑠−𝑥)⁴𝑑𝑠^1/2×𝑛+𝛽𝑒^{∞−𝑛𝑥−ℎ}_𝑘=0𝑔_𝑘^𝑑+1(𝑛𝑥,ℎ)×𝑘!_{()(𝑘+𝛼+1)/(𝑛+𝛽)𝑘+𝛼/(𝑛+𝛽)}𝜂²(𝑠;𝑥)𝑑𝑠^1/2=𝑛+𝛽²𝐾_𝑛^{∗(𝛼,𝛽)}(𝑠−𝑥)⁴;𝑥𝐾_𝑛^{∗(𝛼,𝛽)}𝜂²(𝑠;𝑥);𝑥. (55)
From Lemma 13, we have lim_𝑛→∞𝑛+𝛽²𝐾_𝑛^{∗(𝛼,𝛽)}(𝑠−𝑥)⁴;𝑥=3𝑥². (56) On the other hand, since 𝜂(𝑠;𝑥)∈𝐶[0,𝑎] and lim_𝑠→𝑥𝜂(𝑠;𝑥)=0, then it follows from Theorem 4 that lim_𝑛→∞𝐾_𝑛^{∗(𝛼,𝛽)}𝜂²(𝑠;𝑥);𝑥=𝜂²(𝑥;𝑥)=0. (57) Therefore, we conclude from (55), (56), and (57) that lim_𝑛→∞𝑛+𝛽𝐾_𝑛^{∗(𝛼,𝛽)}(𝑠−𝑥)²𝜂(𝑠;𝑥);𝑥=0 (58) and then, by taking limit as 𝑛→∞ in (52) and using (58), we find lim_𝑛→∞𝐾_𝑛^{∗(𝛼,𝛽)}𝑓;𝑥−𝑓(𝑥)=𝑓^′(𝑥)𝛽𝑥+ℎ(𝑑+1)+2𝛼+12+𝑥𝑓^′′(𝑥)2! (59) which completes the proof.

Remark 15. For 𝛼=𝛽=0, Theorem 14 represents the Voronovskaya type theorem for the operators given by (46) (see [8]).

Remark 16. For ℎ=0, it yields a Voronovskaya type theorem for the Kantorovich-Stancu type of Szasz-Mirakyan operators given by (10).

Remark 17. Getting 𝛼=𝛽=ℎ=0 in Theorem 14 gives the Voronovskaya type result for the Szasz-Mirakyan-Kantorovich operators given by (7).