Quantum Calculus

Frontmatter

1. q-Derivative and h-Derivative

Abstract

As has been mentioned in the introduction, we shall develop two types of quantum calculus, the q-calculus and the h-calculus. We begin with the notion of a quantum differential.

Victor Kac, Pokman Cheung

2. Generalized Taylor’s Formula for Polynomials

Abstract

In the ordinary calculus, a function, f(x) that possesses derivatives of all Orders is analytic at x = a if it can be expressed as a power series about x = a. Taylor’s theorem teils us the power series is

$$ f(x) = \sum\limits_{n = 0}^\infty {f^{(n)} (a)} \frac{{(x - a)^n }} {{n!}}. $$

(2.1)

The Taylor expansion of an analytic function often allows us to extend the definition of the function to a larger and more interesting domain. For example, we can use the Taylor expansion of e ^x to define the exponentials of complex numbers and Square matrices. We would also like to formulate a q-analogue of Taylor’s formula. But before doing so, let us first consider a more general Situation.

Victor Kac, Pokman Cheung

3. q-Analogue of (x − a) n , n an Integer, and q-Derivatives of Binomials

Abstract

As remarked in Chapter 1, D _q is a linear Operator on the space of polynomials. We shall try to apply Theorem 2.1 to D ≡ D _q . We shall need for that the following q-analogue of n!:

$$ [n]! = \left\{ {\begin{array}{*{20}c} {1 if n = 0,} \\ {[n] \times [n - 1] \times \cdots \times [1] if n = if n = 1,2, \ldots .} \\ \end{array} } \right. $$

(3.1)

Victor Kac, Pokman Cheung

4. q-Taylor’s Formula for Polynomials

Abstract

As has been shown in the previous chapter, P _n (x) = (x − a) _q ⁿ /[n]! satisfies the three requirements of Theorem 2.1 with respect to the linear Operator D _q . Therefore, we now obtain the q-version of Taylor’s formula.

Victor Kac, Pokman Cheung

5. Gauss’s Binomial Formula and a Noncommutative Binomial Formula

Abstract

In this chapter we will encounter two binomial formulas involving q-binomial coefficients. Let us first consider an example similar to the one given in the previous chapter.

Victor Kac, Pokman Cheung

6. Properties of q-Binomial Coefficients

Abstract

Let us examine some properties of the q-binomial coefficients, defined by (4.5), with n and j being nonnegative integers and n ≥ j. Because we will recover the ordinary binomial coefficients if we take q → 1, we expect their q-analogues to have similar properties. Firstly, as already remarked in (5.4),

$$ \left[ {\begin{array}{*{20}c} n \\ j \\ \end{array} } \right] = \frac{{[n]!}} {{[j]![n - j]!}} = \left[ {\begin{array}{*{20}c} n \\ {n - j} \\ \end{array} } \right] $$

(6.1)

follows exactly the classical result. However, the correspondence is more subtle for another identity of binomial coefficients, the Pascal rule:

$$ \left( {\begin{array}{*{20}c} n \\ j \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {n - 1} \\ {j - 1} \\ \end{array} } \right) + \left( {\begin{array}{*{20}c} {n - 1} \\ j \\ \end{array} } \right), 1 \leqslant j \leqslant n - 1. $$

Victor Kac, Pokman Cheung

7. q-Binomial Coefficients and Linear Algebra over Finite Fields

Abstract

In this chapter we explain an important combinatorial meaning of the q-binomial coefficients.

Victor Kac, Pokman Cheung

8. q-Taylor’s Formula for Formal Power Series and Heine’s Binomial Formula

Abstract

We now begin to apply what we have learned so far, particularly q-Taylor’s formula (4.1), to study identities involving infinite sums and products. In order to do this, we first have to remark that the generalized Taylor formula (2.2) about a = 0, and hence the q-Taylor formula (4.1) about c = 0, apply not only to polynomials, but also to formal power series. A formal power series, of the form

$$ f(x) = \sum\limits_{k = 0}^\infty {c_k x^k } , $$

may be thought of as a polynomial of infinite degree. It is “formal” because often we do not worry about whether the series converges or not, and we can operate on (for example, differentiate) the series formally. We have to assume a and c to be zero in order to avoid divergence problems. Of course, f(0) = c₀ by definition.

Victor Kac, Pokman Cheung

9. Two Euler’s Identities and Two q-Exponential Functions

Abstract

Now we have two binomial formulas, namely Gauss’s binomial formula (5.5) (with x and a replaced by 1 and x respectively)

$$ (1 + x)_q^n = \sum\limits_{j = 0}^n {q^{j(j - 1)/2} } \left[ {\begin{array}{*{20}c} n \\ j \\ \end{array} } \right]x^j , $$

and Heine’s binomial formula (8.1)

$$ \frac{1} {{(1 - x)_q^n }} = \sum\limits_{j = 0}^\infty {\frac{{[n][n + 1] \cdots [n + j - 1]}} {{[j]!}}} x^j . $$

Victor Kac, Pokman Cheung

10. q-Trigonometric Functions

Abstract

The q-analogues of the sine and cosine functions can be defined in anal-ogy with their well-known Euler expressions in terms of the exponential function.

Victor Kac, Pokman Cheung

11. Jacobi’s Triple Product Identity

Abstract

We recall that the two Euler identities, (9.3) and (9.4), relate infinite products and infinite sums. In this chapter, we will use them to prove an important identity first discovered by Jacobi. Several interesting appli-cations of this identity in number theory will be explored in subsequent chapters.

Victor Kac, Pokman Cheung

12. Classical Partition Function and Euler’s Product Formula

Abstract

With various Substitution of q and z, Jacobi’s triple product identity gives many interesting results. For example, if we put q = q^3/2 and then z = −q^−1/2 into (11.1), we get

$$ \sum\limits_{n \in \mathbb{Z}} {( - 1)^n q^{\frac{{(3n^2 - n)}} {2}} } = \prod\limits_{n = 1}^\infty {(1 - q^{3n} )(1 - q^{3n - 2} )} (1 - q^{3n - 1} ) = \prod\limits_{n = 1}^\infty {(1 - q^n ),} $$

(12.1)

which is called Euler’s product formula. We proved that it holds when |q| < 1. It follows that it also holds as an equality of formal power series in q (see Chapter 8). The formula may also be written using Euler’s product

$$ \phi (q) = \prod\limits_{n = 1}^\infty {(1 - q^n )} $$

as

$$ \phi (q) = \sum\limits_{n \in \mathbb{Z}} {( - 1)^n q^{e_n } } , $$

(12.2)

where

$$ e_n = \frac{{3n^2 - n}} {2} $$

(12.3)

are called pentagonal numbers. The reader is encouraged to multiply out the first few factors of Euler’s product to discover the astonishing fact that indeed the enth coefficient is (−1) ⁿ and all other coefficients are zero.

Victor Kac, Pokman Cheung

13. q-Hypergeometric Functions and Heine’s Formula

Abstract

For further study of infinite sums and infinite products we would like to introduce the hypergeometric series. A classical hypergeometric series is defined as follows.

Victor Kac, Pokman Cheung

14. More on Heine’s Formula and the General Binomial

Abstract

Inspired by (13.16) and (13.17), it is natural to generalize the notion of a q-binomial in the following way.

Victor Kac, Pokman Cheung

15. Ramanujan Product Formula

Abstract

In this chapter, we apply Heine’s formula to prove a remarkable identity discovered by the Indian mathematician Ramanujan. This identity relates a bilateral q-hypergeometric series to an infinite product, and it has many interesting applications in number theory, which will be discussed in subsequent chapters.

Victor Kac, Pokman Cheung

16. Explicit Formulas for Sums of Two and of Four Squares

Abstract

One of the oldest problems in number theory concerns the partition of an integer into a sum of squares. A famous result, first proved by Lagrange, is that any positive integer is a sum of four squares. In this chapter, we will not only prove this theorem, but also will find explicit formulas of Gauss and of Jacobi for the number of partitions of an integer into a sum of two and of four squares.

Victor Kac, Pokman Cheung

17. Explicit Formulas for Sums of Two and of Four Triangular Numbers

Abstract

Besides partitions into square numbers, the Ramanujan product formula can also be applied to the study of partitions into sums of two or four triangular numbers. Let us recall the definition of the nth triangular number, introduced in Chapter 12:

$$ \Delta _n = \frac{{n(n + 1)}} {2}. $$

Victor Kac, Pokman Cheung

18. q-Antiderivative

Abstract

After studying various applications, let us return to q-calculus. So far, we have talked about quantum differentiation only. What about quantum integration? Let us first consider the q-antiderivative.

Victor Kac, Pokman Cheung

19. Jackson Integral

Abstract

Suppose f(x) is an arbitrary function. To construct its q-antiderivative F(x), recall the operator \( \hat M_q \), defined by \( \hat M_q (F(x)) = F(qx) \) in Chapter 5. Then we have by the definition of a q-derivative:

$$ \frac{1} {{(q - 1)x}}(\hat M_q - 1)F(x) = \frac{{F(qx) - F(x)}} {{(q - 1)x}} = f(x). $$

(19.1)

Victor Kac, Pokman Cheung

20. Fundamental Theorem of q-Calculus and Integration by Parts

Abstract

In ordinary calculus, a derivative is defined as the limit of a ratio, and a definite integral is defined as the limit of an infinite sum. Their subtle and surprising relation is given by the Newton-Leibniz formula, also called the fundamental theorem of calculus. In contrast, since the introduction of the definite q-integral has been motivated by an antiderivative, the relation between the q-derivative and definite q-integral is more obvious. Analogous to the ordinary case, we have the following fundamental theorem, or Newton-Leibniz formula, for q-calculus.

Victor Kac, Pokman Cheung

21. q-Gamma and q-Beta Functions

Abstract

Being related to solutions of special types of differential equations, many important functions in analysis are defined in terms of definite integrals.

Victor Kac, Pokman Cheung

22. h-Derivative and h-Integral

Abstract

We have thus far studied only q-calcultus. Now we turn to h-calculus. Firstly, let us recall from Chapter another quantum derivative that is characterized by an additive parameter h, the h-derivative:

$$ D_h f(x) = \frac{{f(x + h) - f(x)}} {h}, $$

where h ≠ 0. Let us begin by developing the properties of h-calculus in an analogous way to what we have done for q-calculus, and discuss its applications in subsequent chapters.

Victor Kac, Pokman Cheung

23. Bernoulli Polynomials and Bernoulli Numbers

Abstract

In this chapter, we introduce a sequence of polynomials that is closely related to the h-antiderivative of polynomials and has many important applications.

Victor Kac, Pokman Cheung

24. Sums of Powers

Abstract

We now turn to the relation between the Bernoulli polynomials and h-calculus. By Proposition 23.1, we have

$$ D_1 B_n (x) = B_n (x + 1) - B_n (x) = nx^{n - 1} , $$

or

$$ n\int {x^{n - 1} d_1 x = B_n (x),} $$

(24.1)

where D₁ is the h-derivative with h = 1 and ∝ f(x)d₁x stands for the h-antiderivative with h=1. Applying the fundamental theorem of h-calculus (22.14), we have for a nonnegative integer n,

$$ a^n + (a + 1)^n + \cdots + (b - 1)^n = \int_a^b {x^n d_1 x = \frac{{B_{n + 1} (b) - B_{n + 1} (a)}} {{n + 1}}} , $$

(24.2)

where a < band b - a ∈ ℤ. If we rewrite the right-hand side using (23.5) and let a = 0, b = M + 1, we get

$$ \sum\limits_{k = 0}^M {k^n } = \frac{1} {{n + 1}}\sum\limits_{j = 0}^n {\left( {_j^{n + 1} } \right)} (M + 1)^{n + 1 - j} b_j . $$

(24.3)

Victor Kac, Pokman Cheung

25. Euler-Maclaurin Formula

Abstract

In q-calculus, the Jackson formula (19.2) provides a way to compute explicitly a q-antiderivative of any function. Recall that the Jackson formula was deduced formally using operators. We will do a similar thing for the h-antiderivative in this chapter.

Victor Kac, Pokman Cheung

26. Symmetric Quantum Calculus

Abstract

The q- and h-differentials may be “symmetrized“ in the following way,

$$ \tilde d_q f(x) = f(qx) - f(q^{ - 1} x), $$

(26.1)

$$ \tilde d_h g(x) = g(x + h) - g(x - h), $$

(26.2)

where as usual, q ≠ 1 and h ≠ 0. The definitions of the corresponding derivatives follow obviously:

$$ \tilde D_q f(x) = \frac{{\tilde d_q f(x)}} {{\tilde d_q x}} = \frac{{f(qx) - f(q^{ - 1} x)}} {{(q - a^{ - 1} )x}}, $$

(26.3)

$$ \tilde d_h g(x) = \frac{{\tilde d_h g(x)}} {{\tilde d_h x}} = \frac{{g(x + h) - g(x - h)}} {{2h}}. $$

(26.4)

We are going to concern ourselves briefly with symmetric q-calculus only, since it is important for the theory of some algebraic objects called quantum groups.

Victor Kac, Pokman Cheung

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