Abstract
In an earlier work, a method was introduced for obtaining indefinite q-integrals of q-special functions from the second-order linear q-difference equations that define them. In this paper, we reformulate the method in terms of q-Riccati equations, which are nonlinear and first order. We derive q-integrals using fragments of these Riccati equations, and here only two specific fragment types are examined in detail. The results presented here are for the q-Airy function, the Ramanujan function, the discrete q-Hermite I and II polynomials, the q-hypergeometric functions, the q-Laguerre polynomials, the Stieltjes-Wigert polynomial, the little q-Legendre and the big q-Legendre polynomials.
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1 Introduction and preliminaries
In [14], we introduced a method to obtain indefinite q-integrals of the form
where the functions p(x) and r(x) are continuous functions in an interval I and the function y(x) is a solution of the second-order q-difference equation
f(x) is a solution of
and h(x) is an arbitrary function. We also introduced
where y(x) is a solution of
F(x) is a solution of
and k(x) is an arbitrary function. The indefinite q-integral
means that \(D_q F(x) = f(x)\), where \(D_q\) is the Jackson’s q-difference operator, which is defined in (1.13) below. The indefinite q-integrals in (1.1) and (1.4) generalize Conway’s indefinite integral
where y(x) is a solution of
f(x) is a solution of \( f'(x)= p(x) f(x) \) and h(x) is an arbitrary function. See [2,3,4,5,6,7, 10]. Conway in [8, 9] reformulated (1.8) to take the form
where
and u(x) is an arbitrary function. Then, he derived many indefinite integrals by considering fragments of the Riccati equation
of the form
or
He identified (1.11) as the Bernoulli fragment, and (1.12) as the linear fragment. This paper is organized as follows. In the remainder of this section, we present the q-notations and concepts required in the next sections. In Sect. 2, we provide a q-analogue of Conway’s indefinite integral formula in (1.10) to the q-setting, along with applications to q-hypergeometric functions, q-Legendre polynomials, discrete q-Hermite I and II polynomials, the q-Airy function, and the Ramanujan function. Section 3 contains applications to the discrete q-Hermite I and II polynomials, the q-Airy function, and the Ramanujan function. In Sect. 4, we introduce new q-integrals by setting \(u(x) = \frac{a}{x}+b\), with appropriate choice of a and b in (6.2) and (6.4). Finally, we added an appendix for all q-special functions, we used in this paper.
Throughout this paper, q is a positive number less than 1, \({\mathbb {N}}\) is the set of positive integers, and \({\mathbb {N}}_0\) is the set of non-negative integers. We use I to denote an interval with zero or infinity as an accumulation point. We follow Gasper and Rahman [11] for the definitions of the q-shifted factorial, q-gamma, q-beta function, and q-hypergeometric series.
A q-natural number \([n]_q\) is defined by \( [n]_q=\frac{1-q^n}{1-q},\,\,n\in {\mathbb {N}}_0\). Jackson’s q-derivative of a function f is denoted by \(D_qf(x)\) and is defined as
provided that \(f'(0)\) exists (see [13,14,15]). Jackson’s q-integral of a function f is defined by
provided that the corresponding series in (1.14) converges, see [16].
The fundamental theorem of q-calculus [1, Eq. (1.29)]
If f is continuous at zero, then
2 q-Integrals from Riccati fragments
In this section, we extend Conway’s result (1.10) to functions satisfying homogenous second-order q-difference equation of the form (1.2) or (1.5). Consider the q-Riccati equations
and
where A(x) and \( \tilde{A}(x)\) are defined as in (2.5) and (2.11), respectively. We can prove that Eqs. (2.1), (2.2) are equivalent to Eqs. (1.2), (1.5) by setting \( \frac{D_q y(x)}{y(x)}= u(x) \) \(\left( \frac{D_{q^{-1}} y(x)}{y(x)}= u(x) \right) \), respectively. This leads to Theorems 2.1 and 2.2 below.
Theorem 2.1
Let y(x) and f(x) be solutions of Eqs. (1.2) and (1.3) in an open interval I, respectively. Let u(x) be a continuous function on I and h(x) be an arbitrary function satisfying
Then,
where the functions p(x), r(x) are defined as in (1.2) and
Proof
Equation (1.1) can be written as
Then, from (2.3), we get
Hence,
Also,
Substituting with (2.7) and (2.8) into (2.6), we get (2.4) and completes the proof. \(\square \)
Theorem 2.2
Let y(x) and F(x) be solutions of Eqs. (1.5) and (1.6) in an open interval I, respectively. Let u(x) be a continuous function on I and k(x) be an arbitrary function satisfying
Then,
where the functions p(x), r(x) are defined as in (1.5) and
Proof
The proof follows similarly as the proof of Theorem 2.1 and is omitted. \(\square \)
The q-integrals presented in the sequel are obtained by choosing the function u(x) to be a solution of a fragment of the q-Riccati equations (2.1) or (2.2). Bernoulli and linear fragments of (2.1) are defined as
respectively. Similarly, the Bernoulli and linear fragments of (2.2) are defined as
and
respectively. The trivial solution \( u(x) =0 \) of (2.12) implies that \( h(x) =c\) is a solution of (2.3), where c is a non-zero constant. Then, (2.4) becomes
Similarly, the trivial solution \( u(x) =0 \) of (2.14) implies that \( k(x) =c\) is a solution of (2.9), where c is a non-zero constant. Then, (2.10) becomes
Theorem 2.3
If g(x) is a solution of the first-order q-difference equation
where A(x) is the function which is defined in (2.5). Then,
is a solution of (2.12) and (2.4) takes the form
Proof
In Theorem 2.1, we choose u(x) to be a solution of (2.12). This produces (2.20). But one can verify that if we set \(u(x)=\frac{1}{v(x)}\), then (2.12) takes the form
which can be rewritten as \(D_{q^{-1}}\Big (\frac{v(x)}{g(x)}\Big )=\frac{1}{g(x/q)} \) or equivalently, \(D_{q}\Big (\frac{v(x)}{g(x)}\Big )=\frac{1}{g(x)} \). Hence, from (1.15), we get \( v(x)= g(x)\int _{0}^{x} \frac{1}{g(t)} \mathrm{{d}}_qt\). Hence, \(u(x)= \frac{1}{v(x)}\) is defined as in (2.19). \(\square \)
Theorem 2.4
Assume that g(x) is defined as in Theorem 2.3 in an interval I containing zero. Then,
Proof
From (2.3),
Hence,
Therefore,
where c is a constant, we can choose \(c=1\). Hence,
\(\square \)
Theorem 2.5
Let I be an interval containing zero. Let p(x) and r(x) be continuous functions at zero. If f(x) is a solution of Eq. (1.3), then
is a solution of Eq. (2.13) in I and (2.4) takes the form
Proof
Multiplying both sides of (2.13) by f(x), we obtain
or equivalently
Hence, from (1.15), we get (2.22). If u(x) is a solution of the q-linear fragment (2.13), then from (2.4), we obtain (2.23) and completes the proof. \(\square \)
3 q-Integrals from the Bernoulli fragment
This section contains indefinite q-integrals that are derived from the q-Bernoulli fragment (2.12).
Theorem 3.1
Proof
By comparing Eq. (A4) with Eq. (1.2), we get \( p(x)=0 \) and \( r(x) = - 1.\) Then, \(f(x) =1\) is a solution of (1.3) and \(g(x) = ( -q(1-q)^2 x^2;q^2)_\infty \) is a solution of (2.18) with \(A(x)= \frac{x}{q}(1-q)\). By Theorem 2.3,
using (A19), we get
By Theorem 2.4,
Substituting with u(x), f(x), and h(x) into (2.20) and using the q-difference equations (A7) and (A8), we get (3.1) and (3.2), respectively. \(\square \)
Theorem 3.2
Let \(\,_2\phi _1(q^a,q^b;q^c;q,x)\) be the q-hypergeometric functions, a, b, and c are real numbers, \(c < 1\), \(\delta > a+b-c\), and \(c\ne q^{-n}\), \(n\in {\mathbb {N}}_0\). Then,
where \(q^\delta =q^a+q^b-q^{a+b} \) and \(\mu = q^{c+1-c(a+b-c)} \).
Proof
By comparing (A17) with Eq. (1.2), we get
Then,
is a solution of (1.3)
satisfies (2.18). By Lemmas 2.3 and A.1, we have
satisfies (2.21). Therefore, by Theorem 2.4, we obtain
By substituting with f(x), h(x), and u(x) into (2.20), we get
where \(B_q (\alpha ,\beta ;x)\) is a function defined in (A18). Using (A18) and (A19), we get the desired result. \(\square \)
Theorem 3.3
If \(y(x)= x^{1-c} \,_2\phi _1(q^{a+1-c},q^{b+1-c};q^{2-c};q,x)\), \(c < 1\), \(\delta > a+b-c\), and \(q^c \ne q^{n+2}\), \(n\in {\mathbb {N}}_0\), is the q-hypergeometric functions. Then,
where \( { \lambda = \dfrac{ [a+1-c]_q[b+1-c]_q}{q^{1-c}[a]_q[b]_q},\,\, \mu =q^{-c(a+b-c-2)}, and \, q^\delta =q^a+q^b\!-q^{a+b} }\).
Proof
By substituting with f(x), h(x), and u(x) as in Theorem 3.2 and \(y(x)= x^{1-c} \,_2\phi _1(q^{a+1-c},q^{b+1-c};q^{2-c};q,x)\) into (2.20) and using (A18) and (A19), we get the desired result. \(\square \)
Theorem 3.4
If \(y(x)= x^{-a} \,_2\phi _1\bigg (q^a,q^{a+1-c};q^{a+1-b};q,\frac{q^{c-a-b+1}}{x}\bigg )\), \(c < 1\), \(\delta > a+b-c\), and \(q^a \ne q^{b-n-1}\), \(n\in {\mathbb {N}}_0\) is the q-hypergeometric functions. Then,
where \({ \lambda = \dfrac{-q^{2-a-b+c} [a+1]_q [a+1-c]_q}{[b]_q[a+1-b]_q},\,\, \mu = q^{(a+1)+c(1-a-b+c)},\,and \, q^\delta }{=q^a+q^b-q^{a+b} }\).
Proof
By substituting with f(x), h(x), and u(x) as in Theorem 3.2 and \(y(x)= x^{-a} \,_2\phi _1\bigg (q^a,q^{a+1-c};q^{a+1-b};q,\frac{q^{c-a-b+1}}{x}\bigg )\) into (2.20) and using (A18) and (A19), we get the desired result. \(\square \)
4 q-Integrals from the linear fragment
In the following results, we obtain new indefinite q-integrals from the linear fragment (2.13).
Theorem 4.1
If \(\mid a\mid < \sqrt{\frac{q}{1-q}}\), then
where \(\sin (x;q)\) and \(\cos (x;q)\) are defined in (A5) and (A6), respectively.
Proof
From (A4), we have \( p(x)=0 \) and \( r(x) = - 1.\) Then, \(f(x) =1\) is a solution of (1.3). By Theorem 2.5, the function \( u(x) =qx \) is a solution of (2.13). Hence,
is a solution of (2.3). Substituting with u(x) and h(x) into (2.23) and using the q-difference equations (A7) and (A8), we get (4.1) and (4.2), respectively. \(\square \)
Theorem 4.2
Let \(n\in {\mathbb {N}}\). If \(p_n(x;-1;q)\) is the big q-Legendre polynomial which is defined in (A22), \(r_n= \frac{2-q^{-n}-q^{n+1}}{1-q}\), then
Proof
By comparing (A23) with (1.2), we get
Then, \(f(x) = (1-x^2)\) is a solution of (1.3). From (2.22), we have
and \( h(x) =\frac{\bigg (x^2;q^2\bigg )_\infty }{\bigg (r_n x^2;q^2\bigg )_\infty }\) is a solution of (2.3). By substituting with u(x) and h(x) into (2.23) and using the q-difference equation
we get (4.4). \(\square \)
Theorem 4.3
Let \(n\in {\mathbb {N}}\). If \( p_n(x|q)\) is the little q-Legendre polynomials defined in (A24), \(r_n=\frac{2-q^{-n}-q^{n+1}}{1-q}\), then
Proof
By comparing Eq. (A25) with (1.2), we get
Then, \(f(x) = x(1-qx)\) is a solution of (1.3). From (2.22), we get \(u(x) = \frac{-q^{1-n}[n]_q[n+1]_q}{(1-qx)}\), h(x) satisfies the q-difference equation (2.3). Consequently, \( h(x)=\frac{(qx;q)_\infty }{(r_nqx;q)_\infty }\). By substituting with u(x) and h(x) into (2.23) and using the q-difference equation
we get (4.6). \(\square \)
5 q-Integrals from arbitrary parts from Riccati equation
In this section, we discuss an approach that chooses u(x) to be a solution of a fragment of the Riccati equation, where a fragment is an equation obtained from Riccati’s equation by deleting one or more of the terms.
Theorem 5.1
Let \(n \in {\mathbb {N}}\) and c be a real number. If \( h_n(x;q)\) is the discrete q-Hermite I polynomial of degree n which is defined in (A9), then
and
Proof
The discrete q-Hermite I polynomial of degree n is defined in (A9) and satisfies the second-order q-difference equation (A10). By comparing (A10) with (1.2), we get
Then,
is a solution of (1.3). Therefore Eq. (2.4) becomes
By taking the fragment
we get
Hence,
is a solution of (2.3). Substituting with the values of h(x) into (5.7) and using
see [17, Eq. (3.28.7)], we get (5.1) for \(c \ne 0\) and (5.2) for \(c =0\). To prove (5.3), we consider the fragment
then \( u(x) = \frac{q^{1-n}}{1-q}x\) and \( h(x) =\dfrac{1}{( q^{1-n}x^2;q^2)_ \infty }\) is a solution of (2.3). Substituting with h(x) and u(x) into (5.7) and using (5.11), we get (5.3). Finally, the proof of (5.4) follows by taking the fragment
In this case, \( u(x) = \frac{[n]_q}{x}\) and \( h(x) = x^n\) is a solution of (2.3). Substituting with h(x) and u(x) into (5.7) and using (5.11), we get (5.4). \(\square \)
Theorem 5.2
Let \(n \in {\mathbb {N}}\) and c be a real number. If \(\widetilde{h}_n(x;q)\) is the discrete q-Hermite II polynomial of degree n which is defined in (A11), then
and
Proof
The discrete q-Hermite II polynomial of degree n is defined in (A11) and satisfies the second-order q-difference equation (A12). By comparing (A12) with (1.5), we get
Then, \( F(x) = \frac{1}{(-x^2;q^2)_\infty }\) is a solution of (1.6), and (2.10) becomes
Consider the fragment
Hence,
and
is a solution of (2.9). Substituting with u(x) and the values of k(x) into (5.16), and using [17, Eq. (3.29.7)] (with x is replaced by \(\frac{x}{q}\) )
we get (5.12) for \(c \ne 0\) and (5.13) for \(c = 0\). The fragment
Then, \(u(x) =\frac{q^n }{1-q_{}}x\) and \( k(x) =(-q^{n+1}x^2;q^2)_\infty \) is a solution of (2.9). Substituting with k(x) into (5.16), we get (5.14). Similarly, to prove (5.15), we consider the fragment
then we obtain \(u(x) =q^{1-n}[n]_q\dfrac{1}{x}\) and \(k(x) =x^n\). Substituting with u(x) and k(x) into (5.16) yields (5.15). \(\square \)
Theorem 5.3
Let c be a real number. If \( Ai_q(x)\) is the q-Airy function which is defined in (A13), then
Proof
The q-Airy function is defined in (A13) and satisfies the second-order q-difference equation (A14). By comparing (A14) with (1.2), we get
By taking the fragment (5.8), we get u(x) and h(x) as in (5.9) and (5.10), respectively. Therefore, (2.4) takes the form
Denote the right hand side of Eq. (5.25) by H(x). I.e
Then, from (1.15), we obtain
From (1.3), we obtain \(f(qx)= -q f(x),\)
then we get
Since
and using the value of h(x) at \(c \ne 0\), we get
Hence, \(\displaystyle \lim _{n \rightarrow \infty } H(q^n x)=0\). From (1.14),(5.9), (5.10) with \(c \ne 0\) and (5.26), we obtain
Combining Eqs. (5.29) and (5.30) yields (5.21). Substituting with the value of \(h(x)=x\) at (\(c = 0\)) yields (5.22). Now, we prove (5.23), by taking the fragment
which implies that \( u(x) = \dfrac{q(1+q)+x}{q(1-q)x}\). Since h(x) satisfies (2.3), then
then we get
Substituting with u(x) into (2.4) and using equations (5.27), (5.28), and (5.31), we get (5.23). \(\square \)
Theorem 5.4
Let \(c \in {\mathbb {R}}\). If \( A_q(x)\) is the Ramanujan function which is defined in (A15), then
Proof
The Ramanujan function is defined in (A15) and satisfies the second-order q-difference equation (A16). By comparing (A16) with (1.2), we get
By taking the fragment (5.8), we get u(x) and h(x) as in (5.9) and (5.10), respectively. Therefore, (2.4) takes the form
Denote the right hand side of Eq. (5.36) by G(x). That is
Then, from (1.15), we get
From (1.3), we obtain \(f(qx)= q^2 x f(x).\) Consequently,
then
Since
substituting with the value of h(x) at \(c \ne 0\), then
Hence, \(\displaystyle \lim _{n \rightarrow \infty } G(q^n x)=0\). From (1.14), (5.9), (5.10) with \(c\ne 0\) and (5.37), we obtain
Combining equations (5.40) and (5.41) yields (5.32). Substituting with the value of \(h(x)=x\) at \(c = 0\) yields (5.33). Now, we prove (5.34), by taking the fragment
which implies that \( u(x) = \frac{x(1+q)-1}{(1-q)x^2}\). Since h(x) satisfies (2.3), then
then we get
Substituting with u(x) into (2.4) and using (5.38), (5.39), and (5.42), we get (5.34). \(\square \)
Theorem 5.5
Let \(n\in {\mathbb {N}}\). The following statements are true:
-
(a)
If \(h_n(x;q)\) is the discrete q-Hermite I polynomial of degree n which is defined in (A9), then
$$\begin{aligned} \int (q^2x^2;q^2)_\infty h_n(x;q) \mathrm{{d}}_qx = -q^{n-1}(1-q)(x^2;q^2)_\infty h_{n-1}\bigg (\frac{x}{q};q\bigg ).\qquad \end{aligned}$$(5.43) -
(b)
If \( p_n(x;a,b;q)\) is the big q-Laguerre polynomial of degree n which is defined in (A26), then
$$\begin{aligned} \int \dfrac{(\frac{x}{a},\frac{x}{b};q)_\infty }{(x;q)_\infty }p_n(x;a,b;q) \mathrm{{d}}_qx = \dfrac{abq^2 (1-q) }{(1-aq)(1-bq)}\dfrac{\bigg (\frac{x}{aq},\frac{x}{bq};q\bigg )_\infty }{(x;q)_\infty } p_{n-1}(x;aq,bq;q).\nonumber \\ \end{aligned}$$(5.44) -
(c)
If \(\alpha > -1\) and \( L_n^{\alpha }(x;q)\) is the q-Laguerre polynomial of degree n which is defined in (A28), then
$$\begin{aligned} \int \dfrac{x^\alpha }{(-x;q)_\infty }L_n^{\alpha }(x;q) \mathrm{{d}}_qx = \dfrac{x^{\alpha +1} }{[n]_q(-x;q)_\infty } L_{n-1}^{\alpha +1}(x;q). \end{aligned}$$(5.45)
Proof
The proof of (a) follows by substituting with r(x) and f(x) from (5.5) and (5.6), respectively, into (2.16). The proof of (b) follows by comparing (A27) with (1.2) to get
Hence, \(f(x) = \dfrac{(\frac{x}{a},\frac{x}{b};q)_\infty }{(qx;q)_\infty }\) is a solution of (1.3). Substituting with r(x) and f(x) into Eq. (2.16) and using
see [17, Eq. (3.11.7)], we get (5.44). To prove (c), compare (A29) with (1.2) to obtain
Hence, \( f(x) = \dfrac{x^{\alpha +1} }{(-qx;q)_\infty }\) is a solution of (1.3). Finally, we prove (5.45) by substituting with r(x) and f(x) into (2.16) and using
see [17, Eq. (3.21.8)]. \(\square \)
Remark 1
-
(a)
The indefinite q-integral (5.44) is nothing else but [14, Eq. (42)] or [17, Eq. (3.11.9)] (with n is replaced by \(n-1\) )
$$\begin{aligned} D_q\left( w(x;aq,bq;q)p_{n-1}(x;aq,bq;q) \right) = \frac{(1-aq)(1-bq)}{abq^2(1-q)}w(x;a,b;q)p_{n}(x;a,b;q), \end{aligned}$$where \( w(x;a,b;q)= \frac{(\frac{x}{b},\frac{x}{a};q)_\infty }{(x;q)_\infty }\).
-
(b)
The indefinite q-integral (5.45) is equivalent to [14, Eq. (46)] (if \(m=n\) ) and to [17, Eq. (3.21.10)] (if \(m=0\)) (with \(\alpha \) is replaced by \(\alpha +1\) and n is replaced by \(n-1\) )
$$\begin{aligned} D_q\left( w(x;\alpha +1;q) L_{n-1}^{\alpha +1}(x;q) \right) = [n]_q w(x;\alpha ;q)L_{n}^{\alpha }(x;q), \end{aligned}$$where \( w(x;\alpha ;q)= \displaystyle {\frac{x^\alpha }{(-x;q)_\infty }}\).
Theorem 5.6
The following statements are true:
-
(a)
If \(\widetilde{h}_n(x;q)\) is the discrete q-Hermite II polynomial of degree n which is defined in (A11), then
$$\begin{aligned} \int \frac{\widetilde{h}_n(x;q)}{(-x^2;q^2)_\infty } \mathrm{{d}}_qx = -\dfrac{q^{1-n}(1-q)}{(-x^2;q^2)_\infty } \widetilde{h}_{n-1}(x;q). \end{aligned}$$(5.48) -
(b)
If \(\nu \) is a real number, \( \nu > -1 \), then
$$\begin{aligned} \int \dfrac{qx^2-q^{1-\nu }[\nu ]_q^2}{x(- x^2 (1-q)^2;q^2)_\infty } J_\nu ^{(2)}( x |q^2) \mathrm{{d}}_qx = \frac{-x}{(- x^2 (1-q)^2;q^2)_\infty } D_{q^{-1}} J_\nu ^{(2)}( x |q^2). \end{aligned}$$
Proof
The proof of (a) follows by substituting with r(x) and F(x) as in the proof of Theorems (5.2) into Eq. (2.17). To prove (b), compare (A21) with (1.5) to obtain
Hence, \( F(x) = \dfrac{x}{(- x^2 \lambda ^2(1-q)^2;q^2)_\infty }\) is a solution of (1.6). Substituting with r(x) and F(x) into Eq. (2.17). \(\square \)
Remark 2
The indefinite q-integral (5.48) is equivalent to [14, Eq. (68)] and to
where \( w(x;q)= \frac{1}{(-x^2;q^2)}\), see [17, Eq. (3.29.9)].
Theorem 5.7
The following statements are true:
-
(a)
If \( Ai_q(x)\) is the q-Airy function which is defined in (A13), then
$$\begin{aligned} \sum _{k=0}^{\infty } (-q)^k Ai_q(q^kx) = \frac{1}{1+q}\,_1\phi _1 (0;-q^2;q,-x). \end{aligned}$$(5.49) -
(b)
If \( A_q(x)\) is the Ramanujan function which is defined in (A15), then
$$\begin{aligned} \sum _{k=0}^{\infty } q^{\frac{k(k+1)}{2}}x^k A_q(q^k x) = - \,_0\phi _1 (-;0;q,-q^2x). \end{aligned}$$(5.50) -
(c)
If \(S_n(x;q)\) is the Stieltjes–Wigert polynomial of degree n \((n \in {\mathbb {N}})\) which is defined in (A30), then
$$\begin{aligned} \sum _{k=0}^{\infty } q^{\frac{k(k+1)}{2}}x^k S_n(q^kx;q)=\frac{1}{1-q^n} S_{n-1}(qx;q). \end{aligned}$$(5.51)
Proof
The proof of (a) follows by substituting with r(x) from (5.24) into (2.16) and using (5.27) and (5.28). The proof of (b) follows by substituting with r(x) from (5.35) into (2.16) and using (5.38) and (5.39). To prove (c), compare Eq. (A31) with (1.2) to get
From (1.3), we obtain \(f(qx)= q^2 x f(x).\) Consequently,
then
Substituting with r(x) into (2.16) and using (1.14), (5.52), and
6 q-Integrals from substitution of simple algebraic forms
In this section, we substitute into Eq. (2.4) with simple algebraic forms for u(x) which involve arbitrary constants, such as
to derive indefinite q-integrals. Set
Then, (2.4) will be
where the constants a and b in Eq. (6.1) are chosen so that \(S_q(x)\) has a simple form. Also, we define
Then, (2.10) will be
Theorem 6.1
Let \(n \in {\mathbb {N}}\), \(n\ge 2\). Let \(h_n(x;q)\) be the discrete q-Hermite I polynomial of degree n which is defined in (A9). Then,
and
Proof
From (A10),
Hence, \(f(x) = (q^2x^2;q^2)_\infty \) is a solution of Eq. (1.3). Set u(x) as in (6.1). Then,
If \({ a=1}\) and \(b=0\) in (6.8), then
and \(h(x) = x\) is a solution of (2.3). By substituting with u(x), \(S_q(x)\), and h(x) into (6.3) and using (5.11), we get (6.6). If \(a=[n]_q\) and \(b=0\) in (6.8), then
Hence, \(h(x) = x^n\) is a solution of (2.3). Substituting with u(x), \(S_q(x)\), and h(x) into (6.3) and using (5.11), we get (6.7). \(\square \)
Remark 3
The indefinite q-integral (6.7) is equivalent to (5.4) in Theorem 5.1.
Theorem 6.2
Let \(n \in {\mathbb {N}}\), \(n\ge 2\). Let \(\widetilde{h}_n(x;q)\) be the discrete q-Hermite II polynomial of degree n which is defined in (A11). Then,
and
Proof
From (A12),
Then, \( F(x) = \frac{1}{(-x^2;q^2)_\infty }\) is a solution of (1.6). Set u(x) as in (6.1), we get
If \(a=1\) and \(b=0\) in (6.11), then
Therefore, \(k(x) = x\) is a solution of (2.9). By substituting with u(x), \(T_q(x)\), and k(x) into (6.5) and using Eq. (5.20), we get (6.9). If \(a=q^{1-n} [n]_q\) and \(b=0\) in (6.11), then
Therefore, \(k(x) = x^n\) is a solution of (2.9). By substituting with u(x), \(T_q(x)\), and k(x) into (6.5) and using (5.20), we get (6.10). \(\square \)
Remark 4
The indefinite q-integral (6.10) is equivalent to [14, Eq. (69)] (if \(m=n\)) and to (5.15) in Theorem 5.2.
Theorem 6.3
Let \(n \in {\mathbb {N}}\). If \(S_n(x;q)\) is the Stieltjes-Wigert polynomial of degree n defined in (A30), then
and
Proof
By comparing (A31) with (1.2), we get
Set u(x) as in (6.1). Then,
We set \(a = [n]_q \) and \(b=0\) in (6.14) then
Therefore, \(h(x) = x^n\) is a solution of (2.3). By substituting with h(x), \(S_q(x)\), and u(x) into (6.3), using (5.52) and (5.53), we get (6.12).
If \(a = 1 \) and \(b=0\) in (6.14), then
Therefore, \(h(x) = x\) ia a solution of (2.3). By substituting with h(x), \(S_q(x)\), and u(x) into (6.3) and using (5.52) and (5.53), we get (6.13). \(\square \)
7 Conclusions
A method of deriving q-integrals using fragments of q-Riccati equations has been presented. The method of fragmentation used is analogous to but not equivalent to that presented in [14]. Only two q-Riccati fragments have been presented here in detail, and these give the quadrature formulas presented in Eqs. (2.20) and (2.22)–(2.23).
References
Annaby, M.H., Mansour, Z.S.: q-Fractional Calculus and Equations. Lecture Notes in Mathematics, vol. 2056. Springer, Berlin (2012)
Conway, J.T.: Indefinite integrals of some special functions from a new method. Integr. Transform Spec. Funct. 26(11), 845–858 (2015)
Conway, J.T.: A Lagrangian method for deriving new indefinite integrals of special functions. Integr. Transform Spec. Funct. 26(10), 812–824 (2015)
Conway, J.T.: Indefinite integrals involving the incomplete elliptic integrals of the first and second kinds. Integr. Transform Spec. Funct. 27(5), 371–384 (2016)
Conway, J.T.: Indefinite integrals of Lommel functions from an inhomogeneous Euler–Lagrange method. Integr. Transform Spec. Funct. 27(3), 197–212 (2016)
Conway, J.T.: Indefinite integrals of quotients of special functions. Integr. Transform Spec. Funct. 29(4), 269–283 (2018)
Conway, J.T.: Indefinite integrals of special functions from inhomogeneous differential equations. Integ. Transform Spec. Funct. 30(3), 166–180 (2018)
Conway, J.T.: New indefinite integrals from a method using Riccati equations. Integr. Transform Spec. Funct. 29(12), 927–941 (2018)
Conway, J.T.: More indefinite integrals from Riccati equations. Integr. Transform Spec. Funct. 30(12), 1004–1017 (2019)
Conway, J.T.: Indefinite integrals of special functions from hybrid equations. Integr. Transform Spec. Funct. 31(4), 253–267 (2020)
Gasper, G., Rahman, M.: Basic Hypergeometric Series. Cambridge university Press, Cambridge (2004)
Hahn, W.: Beiträge zur Theorie der Heineschen Reihen. Math. Nachr. 2, 340–379 (1949)
Heine, E.: Handbuch der Kugelfunctionen, Theorie und Anwendungen, vol. 1. G. Reimer, Berlin (1878)
Heragy, G.E., Mansour, Z.S., Oraby, K.M.: An efficient method for indefinite \(q\)-integrals. Prog. Fract. Differ. Appl. 9(3), 1–29 (2023). https://doi.org/10.18576/pfda/090309
Jackson, F.H.: On \(q\)-functions and a certain difference operator. Trans. R. Soc. Edinb. 46, 64–72 (1908)
Jackson, F.H.: On \(q\)-definite integrals. Q. J. Pure Appl. Math. 41, 193–203 (1910)
Koekoek, R., Rene, F.: Swarttouw. The Askey-scheme of hypergeometric orthogonal polynomials and its \(q\)-analog. In: Reports of the faculty of Technical Mathematics and Informatics, No. 98-17, Delft (1998)
Rahman, M.: A note on the orthogonality of Jackson’s \(q\)-Bessel functions. Cand. Math. Bull. 32, 369–376 (1989)
Ohyama, Y.: A unified approach to \(q\)-special functions of the Laplace type. http://arxiv.org/abs/1103.5232
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Appendix A: q-Functions
Appendix A: q-Functions
Jackson introduced three q-analogues of Bessel functions [11, 16], they are defined by
The solutions of the second-order q-difference equation, see [1],
under the initial conditions
are the functions \(\sin (x;q)\) and \(\cos (x;q)\), respectively. The functions \(\sin (z;q)\) and \(\cos (z;q)\) are defined for \( z \in {\mathbb {C}}\) by
The q-trigonometric functions satisfy the q-difference equations
The discrete q-Hermite I polynomial of degree n
satisfies the second-order q-difference equation, see [17, Eq. (3.28.5)],
The discrete q-Hermite II polynomials of degree n
satisfies the second-order q-difference equation, see [17, Eq. (3.29.5)],
The q-Airy function
satisfies the second-order q-difference equation, see [19, Eq. (4)],
The Ramanujan function
satisfies the second-order q-difference equation, see [19, Eq. (5)],
The q-hypergeometric series \(\,_r\phi _s\) is defined by
whenever the series converges, see [11].
The q-hypergeometric functions \( _2\phi _1(q^a,q^b;q^c;q,x)\) satisfy the second-order q-difference equation [11]
The functions
and
are solutions of the basic hypergeometric q-difference Eq. (A17), see [11].
Lemma A.1
Let \(\alpha \) and \(\beta \) be complex numbers with positive real parts. Then,
where \((qt;q)_{\beta -1}=\dfrac{(qt;q)_{\infty }}{(q^{\beta }t;q)_{\infty }} \).
Proof
From (1.14),
Since \((a;q)_n = \dfrac{(a;q)_\infty }{(aq^n;q)_\infty },\) then
\(\square \)
It is worth noting that from Lemma A.1,
One of Heine’s transformations of \(\,_2\phi _1\) series
see [11, Eq. (III.1)]. The second Jackson q-Bessel function
satisfies the second-order q-difference equation [18]
The big q-Legendre polynomials
satisfy the second-order q-difference equation, see [17, Eq. (3.5.17)],
The little q-Legendre polynomials
satisfy the second-order q-difference equation, see [17, Eq. (3.12.16)],
The big q-Laguerre polynomial
satisfies the second-order q-difference equation, see [17, Eq. (3.11.5)],
The q-Laguerre polynomial of degree n
satisfies the second-order q-difference equation, see [17, Eq. (3.21.6)],
The Stieltjes–Wigert polynomials
satisfy the second-order q-difference equation, see [17, Eq. (3.27.5)],
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Heragy, G.E., Mansour, Z.S.I. & Oraby, K.M. Indefinite q-integrals from a method using q-Riccati equations. Ramanujan J 64, 881–914 (2024). https://doi.org/10.1007/s11139-024-00855-0
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DOI: https://doi.org/10.1007/s11139-024-00855-0
Keywords
- q-integrals
- q-Bernoulli fragment
- q-Linear fragment
- Simple algebraic form
- q-Airy function
- Ramanujan function