R E P R E S E N T A T I O N S

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1 Z E S Z Y T Y N A U K O W E A K A D E M I I M A R Y N A R K I W O J E N N E J S C I E N T I F I C J O U R N A L O F P O L I S H N A V A L A C A D E M Y 2017 (LVIII) 4 (211) DOI: / H u b e r t W y s o c k i J A C K S O N A N D H A H N D I F F E R E N C E M O D E L S AS DI SCRETE BITTNER OPERATIONAL CALCULUS R E P R E S E N T A T I O N S ABSTRACT In the paper, there have been constructed discrete models of the non-classical Bittner operational calculus that are related to the notions of Jackson and Hahn derivatives, both known from the quantum calculus. To achieve it, we have used the Bittner operational calculus representation for the forward difference. Key words: operational calculus, derivative, integrals, limit conditions, Jackson difference, Hahn difference. INTRODUCTION Let be a real function determined on the interval. Moreover, let The essence of the -calculus (quantum calculus) [2, 3, 8, 11] is to substitute the shift with a dilatation in the difference quotient Then, we have Polish Naval Academy, Faculty of Mechanical and Electrical Engineering, Śmidowicza 69 Str., Gdynia, Poland; 61

2 Hubert Wysocki The operations and are called an -derivative and a -derivative (or Jackson derivative) [10, 11], respectively. Using an increment we determine the operation called an -derivative (or Hahn derivative) [7, 9]. It is a generalization of and (which are obtained with and, respectively). In the finite difference calculus, a forward difference (1) corresponds to the increment where and is the set of naturals. The -order forward difference where is a given natural number, is a generalization of the operation (1). A MODEL OF THE OPERATIONAL CALCULUS WITH THE TH-ORDER FORWARD DIFFERENCE Let be a set of complex numbers, while a linear space of complex sequences 1 with a usual sequences addition and multiplication of sequences by complexes. 1 Formally, in the operational calculus we differentiate a function symbol from a symbol of a function value at a point. In particular, signifies a sequence, whereas its value for a given. This notation is derived from J. Mikusiński [12]. In what follows, we shall skip the brackets whenever it does not cause ambiguity. 62 Zeszyty Naukowe AMW Scientific Journal of PNA

3 Jackson and Hahn difference models as discrete Bittner operational calculus representations Moreover, let be roots of unity, i.e. where and is the imaginary unit. In [13] it has been shown that the operations, determined on as follows, where (2) (3) (4) form a discrete model (representation) of the so-called Bittner operational calculus and satisfy two basic formulas of this calculus, i.e. (5) FUNDAMENTALS OF THE BITTNER OPERATIONAL CALCULUS The Bittner operational calculus [4 6] is a system in which and are linear spaces (over the same scalar field ) such that. The linear operation (denoted as ), called the (abstract) derivative, is a surjection. Moreover, is a set of indexes for the operations and such that and. and are called integrals and limit conditions, respectively. The kernel of, i.e. is a set of constants for the derivative. The limit conditions are projections of on the subspace. If we define the objects (6), then we speak of a representation or a model of the operational calculus. 4 (211) (6)

4 Hubert Wysocki A MODEL OF THE OPERATIONAL CALCULUS WITH THE TH-ORDER JACKSON DIFFERENCE A set will be called the Jackson time scale (cf. [1]). As previously, let be a linear space of complex sequences with a usual sequences addition and sequences multiplication by complexes. Then. Indeed, each term of a sequence can be presented in the form of where. For, in turn, we have where and. On the space we define a forward -difference (7) It corresponds to an increment which determines the Jackson -derivative. The below -order Jackson difference (8) where is a given natural number, is a generalization of the operation (7). Hence, we have Thus, to the operation, determined on the Jackson time scale, there corresponds the difference defined on the scale. 64 Zeszyty Naukowe AMW Scientific Journal of PNA

5 Jackson and Hahn difference models as discrete Bittner operational calculus representations Therefore, on the basis of (2) (4), to the derivative understood as the forward -difference (8), there correspond integrals (9) and limit conditions (10) where and. For the above operations (9) and (10) the formulas (5) are satisfied. So, we get the Corollary 1. The operations (8) (10) form a discrete Jackson model of the Bittner operational calculus Using Mathematica, we can easily determine the consecutive terms of the sequences (8) (10). For example, for and we obtain equation Example 1. Using the described model, we will solve the -difference (11) with initial conditions (12) Hence we have. 4 (211)

6 Hubert Wysocki The equation (11) can be shown in the form of (13) where. Using Mathematica, on the basis of the formula (10), we determine the limit condition corresponding to the initial conditions (12). Finally, we get (14) The problem (13), (14) has exactly one solution (Th. 3 [6]) Hence, using Mathematica and basing on (9), we obtain a solution to the Cauchy problem (11), (12): A MODEL OF THE OPERATIONAL CALCULUS WITH THE TH-ORDER HAHN DIFFERENCE In the quantum calculus, there are determined the so-called ( -analogs of non-negative integers) -numbers The set i.e., will be called the Hahn time scale (cf. [1]). The below operation (15) which is determined on the space and where and are given, will be called the -order Hahn difference. 66 Zeszyty Naukowe AMW Scientific Journal of PNA

7 Jackson and Hahn difference models as discrete Bittner operational calculus representations Notice that Thus which means that to the operation determined on the Hahn time scale, there corresponds the difference defined on the scale. under- Hence, on the basis of (2) (4), we infer that to the derivative stood as the Hahn difference (15), there correspond the below integrals (16) and limit conditions (17) where and Thus, we come to the Corollary 2. The operations (15) (17) form a discrete Hahn model of the Bittner operational calculus Example 2. In the considered model, using Mathematica, we will determine such a solution to the equation that satisfies the initial conditions (18) We have here as well as the equation (18) in the form of, where. 4 (211)

8 Hubert Wysocki Similarly as before, from the formula we eventually get where denotes the binary logarithm. SOME GENERALIZATIONS In [13], there was also considered a forward difference with the basis which is a generalization of the operation (2). It was shown that to the derivative there correspond the integrals and the limit conditions where, while and are the operations (3) and (4), respectively. Therefore, on the basis of the foregoing considerations we obtain the two following corollaries: Corollary 3. The operations 68 Zeszyty Naukowe AMW Scientific Journal of PNA

9 Jackson and Hahn difference models as discrete Bittner operational calculus representations where, form a discrete Jackson model of the Bittner operational calculus Corollary 4. The operations where operational calculus, form a discrete Hahn model of the Bittner REFERENCES [1] Aldwoah K. A., Hamza A. E., Generalized time scales, Advances in Dynamical Systems and Applications, 2011, 6 (2), pp [2] Annaby M. H., Mansour Z. S., q-fractional Calculus and Equations, Springer, Heilderberg [3] Aral A., Gupta V., Agarwal R. P., Applications of q-calculus in Operator Theory, Springer, New York [4] Bittner R., Operational calculus in linear spaces, Studia Math., 1961, 20, pp [5] Bittner R., Algebraic and analytic properties of solutions of abstract differential equations, Rozprawy Matematyczne [ Dissertationes Math. ], 41, PWN, Warszawa [6] Bittner R., Rachunek operatorów w przestrzeniach liniowych, PWN, Warszawa 1974 [Operational Calculus in Linear Spaces available in Polish]. [7] Brito da Cruz A. M. C., Martins N., Torres D. F. M., Higher-order Hahn s quantum variational calculus, Nonlinear Anal. 2012, 75 (3), pp [8] Ernst T., A Comprehensive Treatment of q-calculus, Birkhäuser/Springer, Basel [9] Hahn W., Über Orthogonalpolynome, die q-differenzengleichungen genügen, Mathematische Nachr., 1949, 2 (1 2), pp [10] Jackson F. H., On q-functions and a certain difference operator, Transactions of the Royal Society of Edinburgh, 1909, 46 (2), pp [11] Kac V., Cheung P., Quantum Calculus, Springer, New York (211)

10 Hubert Wysocki [12] Mikusiński J., Operational Calculus, Pergamon Press, New York [13] Wysocki H., An operational calculus model for the nth-order forward difference, Zeszyty Naukowe Akademii Marynarki Wojennej [ Scientific Journal of Polish Naval Academy ], 2017, No. 3, pp R Ó Ż N I C O W E M O D E L E J A C K S O N A I H A H N A J A K O D Y S K R E T N E R E P R E Z E N T A C J E R A C H U N K U O P E R A T O R Ó W B I T T N E R A STRESZCZENIE W artykule skonstruowano dyskretne modele nieklasycznego rachunku operatorów Bittnera związane z pojęciami pochodnej Jacksona i pochodnej Hahna znanymi z rachunku kwantowego. Do tego celu wykorzystano reprezentację rachunku operatorów Bittnera dla różnicy progresywnej. Słowa kluczowe: rachunek operatorów, pochodna, pierwotne, warunki graniczne, różnica Jacksona, różnica Hahna. 70 Zeszyty Naukowe AMW Scientific Journal of PNA