Discrete Calculus of Sequences

We showed that is possible to define a derivative and a integral of a sequence which satisfy similar proprieties of the derivative and integral of the usual differential and integral calculus. We also study convexity of sequences that can be defined as usual by the second derivative. 1 A little introduction on calculus of finite differences We will introduce some definitions and notation from the calculus of finite differences. Mostly all notation are from Jordan [2], and, like the notation, mostly definitions on the subject are the same, as we can find in Jordan’s [2] book or any other book on finite differences, like Boyle [1]. The definition of symbol from Boyle [1, pp. 16–18] will be generalized, and it will be used later on the study of discrete calculus of sequences. 1.1 Difference The main definition of the calculus of finite differences is the difference. The difference of a function f(x), which is given for x1, x2, . . . , xn; such that xi+1 − xi = h for all i between 1 and n− 1, is ∆ x,h f = f(x+ h)− f(x). 1.2 Operation of displacement The displacement operation is an important operation in calculus of finite differences, and consist in increasing the argument of the function by some amount. Then, if we denote this operation by E, we have E h f(x) = f(x+ h). (1) 1 Note that, in E we are omitting h and x as we will do most time with such operations, and as is vastly done in literature, like in Jordan’s book [2, p. 6], let us define E n by E f(x) = E[Ef(x)] = f(x+ nh), where n are a positive integer, and for a negative integer −n, we have E f(x) = f(x− nh). 1.3 Operation of the mean The operation of the mean will be more important later in this paper than displacement operation, because it is invariant on the interchange of E and 1. This operation will be denoted by M, and is defined as follows: M h f(x) = f(x) + f(x+ h) 2 . (2) 1.4 Symbolic Calculus Let O be the set of all operation already defined and also contains more one operations denoted by 1. Where 1 is the identity operation which take a function to itself. We note that any operation from O are linear and commute with each others. We can define addiction of operation as [A+B]f(x) = Af(x) +Bf(x), for any A and B, and multiplication of a operation by a real number, in this case λ, [λA]f(x) = λAf(x), as well as multiplication of operation that for consistency must be defined as ABf(x) = A[Bf(x)]. Then we see that for operation in O some proprieties such as associativity, distribution of multiplication over addiction, linearity and commutativity are satisfied, moreover is easy to see that for addiction the order of the operations does not matter. Definition 1. Let O be a set of unary operations O : F → F . Where F is any set of function. We say that O is a symbolic set over F if all proprieties below are satisfied (i) Linearity: If S ∈ O. Then S(λf + g) = λSf + Sg, where f and g belong to F , and λ is a real number. (ii) Commutativity (multiplication): For any S and O in O. We have SO = OS. (iii) Commutativity (addiction): Given S and O in O. Then S+O = O+ S. 2 (iv) Associativity: If S, O and G are elements of O. Hence S+ (O+G) = (S+O) +G. (v) Distribution over addiction: Let S, G and O be in O. We must have that S(G+O) = SG+ SO. It is easy to prove that O is a symbolic set over the set of all continuous functions. If S belongs to a symbolic set O, then S is symbol under O. But, S still can be a symbol under O even if S does not belongs to O. Definition 2. Let O be any symbolic set over F , and let S by any unitary operation S : F → F . Hence, S is a symbol under O over F if the set O = O ∪ {S } is also a symbolic set over F . We already state that O is a symbolic set over continuous functions, and as any differentiable function is continuous we have that O is a symbolic set over differentiable functions too. If we consider the differentiation operation denoted by D we can show that it is a symbol under O, obviously over differentiable functions. Definition 3. Let D denote the discrete differentiation operation that act in a function as follows: D h f(x) = f(x+ h)− f(x) h . First, we see that lim h→0 D h = D, which given as a hint about proprieties and interpretation of D. Second, by the definition is easy to see that D is a symbol under O, because D is the product of the symbol ∆ h f , and h which act as a real. A equivalent operation is mentioned by Boyle [1, p. 3] given by the ratio ∆ h ∆x , including that D 1 is equal to ∆ 1 . Remark 4. It very important to remember that D act on any differentiable function, on the other hand D h act in a wider set of function including differentiable function and discrete function such as sequences. 1.5 Relation between symbols under O Now is clear that with respect to addiction, subtraction and multiplication, as we defined for any operation, the symbols works as algebraic quantities. Then we can find relation between symbols, and obtain more symbols by addiction and multiplication of symbols. Usually we will omit the symbolic set when it does not cause any ambiguity.