Abstract
In this paper, we introduce the polynomials \(B^{(k)}_{n,\alpha }(x;q)\) generated by a function including Jackson q-Bessel functions \(J^{(k)}_{\alpha }(x;q)\) \( (k=1,2,3),\,\alpha >-1\). The cases \(\alpha =\pm \frac{1}{2}\) are the q-analogs of Bernoulli and Euler\(^{,}\)s polynomials introduced by Ismail and Mansour for \((k=1,2)\), Mansour and Al-Towalib for \((k=3)\). We study the main properties of these polynomials, their large n degree asymptotics and give their connection coefficients with the q-Laguerre polynomials and little q-Legendre polynomials.
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1 Introduction and Preliminaries
The Bernoulli polynomials \( \left( B_{n}(x)\right) _{n}\) are defined by the generating function
In a series of papers, Frappier [9,10,11] studied the generalized Bernoulli polynomials \( B_{n,\alpha }(x)\), defined by the generating function
where
\( J_{\alpha }(t)\) is the Bessel function of the first kind of order \(\alpha ,\) and \(j_{1,\alpha } \) is the smallest positive zero of \(J_{\alpha }(t)\). Ismail and Mansour, see [19], introduced a pair of q-analogs of the Bernoulli polynomials by the generating functions
They also defined a pair of q-analogs of the Euler polynomials by the generating functions
where
and \(a\in {\mathbb {C}}\), see [12]. The functions \(E_q(x)\) and \(e_q(x)\) are the q-analogs of the exponential functions defined by
see [12].
In [22], Mansour and Al-Towalib introduced q-analogs of Bernoulli and Euler polynomials by the generating functions
where
is a q-analog of the exponential function. This q-exponential function has the property \(\lim _{q\rightarrow 1}\exp _{q}(x) = e^{x}\) for \(x \in {\mathbb {C}}\). It is an entire function of x of order zero, see [12, Eq. (1.3.27), p. 12].
In this paper, we use \({\mathbb {N}} \) to denote the set of positive integers and \({\mathbb {N}}_{0} \) to denote the set of non-negative integers. Throughout this paper, unless otherwise is stated, q is a positive number that is less than one. We follow Gasper and Rahman [12] to define the q-shifted factorial, the q-binomial coefficients, and the q-gamma function. The q-integer number \([n]_{q}\) is defined by
Jackson in [20] defined the q-difference operator by
The symmetric q-difference operator is defined by, see [8, 12],
The q-trigonometric functions \(\sin _{q}z,\, \cos _{q}z,\,Sin_{q}z\) and \(Cos_{q} z\) are defined by
see [5, 12]. The q-sine and cosine functions \(S_{q}(z),\,\, C_{q}(z)\) are defined by the q-Euler formula
where
cf. [8, p. 2]. The hyperbolic functions \(Sh_{q}(z)\) and \(Ch_{q}(z)\) are defined for \(z\in {\mathbb {C}}\) by
There are three known q-analogs of the Bessel function that are due to Jackson [20]. These are denoted by \( J^{(k)}_{\alpha }(t;q)\, (k=1,2,3)\) and defined by
For convenience, we set
The functions \({\mathcal {J}}_{\alpha }^{(k)} (t;q)\, (k=1,2,3)\) are called the modified Jackson q-Bessel functions. From now on, we use \((j^{(k)}_{m,\alpha })_{m=1}^{\infty }\) to denote the positive zeros of \(J^{(k)}_{\alpha }(\cdot ;q^{2})\) arranged in increasing order of magnitude. Consequently, \(j^{(k)}_{1,\alpha }\) is the smallest positive zero of \(J^{(k)}_{\alpha }(\cdot ;q^{2})\,( k=1,2,3)\).
This paper is organized as follows. In Sect. 2, we introduce three q-analogs of the generalized Bernoulli polynomials defined in (1.1). The generating functions of these q-analogs include the three q-analogs of Jackson q-Bessel functions mentioned above. We also include the main properties of these q-analogs. Section 3 introduces a q-Fourier expansion for the generalized Bernoulli numbers related to the first and second Jackson q-Bessel functions. Also, their large n degree asymptotic is derived. Finally, in Sect. 4 as an application, we introduce the connection coefficients between q-analogs and certain q-orthogonal polynomials.
2 Generalized q-Bernoulli Polynomials Generated by Jackson q-Bessel Functions
This section introduces three q-analogs of the generalized Bernoulli polynomials introduced by Frappier in [9,10,11].
Definition 2.1
The generalized q-Bernoulli polynomials \(B^{(k)}_{n,\alpha }(x;q)\,( k=1,2,3)\) are defined by the generating functions
where \(g^{(k)}_{\alpha }(t;q) \,( k=1,2,3 )\) are the functions defined for \((k=1,2)\) by
and
Since the generating functions in (1.2), (1.3), and (1.5) can be written as
and
then, if we substitute with \(\alpha =\pm \frac{1}{2}\) in (2.1), (2.2), and (2.3), we obtain the q-Bernoulli and Euler polynomials defined in (2.4), (2.5) and (2.6), respectively.
Lemma 2.2
For \(n\in {\mathbb {N}}_{0}\) and \(Re \,\alpha >-1\),
Proof
Hahn in [14] proved the identity
Since
then, substituting from (2.8) into (2.7), we conclude that
Hence
which completes the proof. \(\square \)
Definition 2.3
The generalized q-Bernoulli numbers \({\beta }_{n,\alpha }(q),\) \( \beta ^{(3)}_{n,\alpha }(q)\) are defined respectively in terms of the generating functions
Proposition 2.4
For \(n\in {\mathbb {N}}\), we have
Proof
If we substitute with \(x=\frac{1}{2}\) in Eqs. (2.1)–(2.3), we find that their left hand side are even functions. Therefore, the coefficients of the odd powers of \(t^{n}\) on the right hand sides of Eqs. (2.1)–(2.3) vanish. This proves the proposition. \(\square \)
Proposition 2.5
For \( k\in \{1,2,3\}\) and \(n\in {\mathbb {N}}\), the polynomials \(B_{n,\alpha }^{(k)}(x;q)\) have the representation \(B_{0,\alpha }^{(k)}(x;q)=1\),
Proof
We prove the case \((k=1)\). The proofs for \((k=2,3)\) are similar and are omitted. Substituting with the series representation of \(e_{q}(x)\) from (1.4) into (2.1) gives
Hence
where we applied the Cauchy product formula. Equating the nth power of t in (2.15), we obtain (2.12). \(\square \)
Proposition 2.6
For \(n\in {\mathbb {N}}\) and \( k\in \{1,2,3\}\), the polynomials \(B_{n,\alpha }^{(k)}(x;q)\) satisfy the q-difference equations
Proof
We only prove the case (\(k=1\)) and the proofs of \((k=2,3)\) are similar. Calculating the q-derivative of both sides of (2.1) with respect to the variable x and taking into consideration that
we obtain
Therefore,
Equating the corresponding nth power of t in (2.19), we obtain (2.16). \(\square \)
Corollary 2.7
Let \(n\in {\mathbb {N}}\) and k be a positive integer such that \(k\le n\). Then for \(x\in {\mathbb {C}},\)
Proof
The proofs follow from Proposition 2.6 and the mathematical induction. \(\square \)
Proposition 2.8
For \(\mid t\mid <\frac{1}{1-q}\min \{ j^{(1)}_{1,\alpha }, j^{(2)}_{1,\alpha },2\}\),
Proof
Set \( x=\frac{1}{2} \) in (2.1), we obtain
Substituting from (2.9) into (2.22), we obtain (2.20). Similarly, we can prove (2.21). \(\square \)
The following Lemma from [22] gives the reciprocal of \( \exp _{q}(z)\) in a certain domain.
Lemma 2.9
Let \(z\in \Omega , \,\displaystyle \Omega := \{ z\in {\mathbb {C}}: \, \mid 1-\exp _{q}(-z) \mid < 1\}\). Then
where
Proposition 2.10
For \( Re\,\alpha >-1 \) and \(t\in \displaystyle \Omega = \{ t\in {\mathbb {C}}: \, \mid 1-\exp _{q}(-t)\mid < 1\}\),
where \(c_{n}\) is defined in (2.23).
Proof
Substitute with \( x=0\) in Eq. (2.3). This gives
From Lemma 2.9,
Applying the Cauchy product formula, we obtain (2.24) and completes the proof. \(\square \)
Theorem 2.11
For \( n\in {\mathbb {N}}_{0} \) and \(x\in {\mathbb {C}}\),
Proof
If we replace x by \( -x \) in (2.1), then
Since \( e_{q}(-xt)E_{q}(xt)=1,\) then multiplying (2.2) by (2.25) gives
From (2.10), we obtain
Hence
So, equating the nth power of t in (2.26), we obtain the required result. \(\square \)
Proposition 2.12
For \( n\in {\mathbb {N}}_{0},\) \( x\in {\mathbb {C}}\) and \(q\ne 0, \)
In particular,
Proof
Replacing q by \(\frac{1}{q}\) on the generating function in (2.1) and using \( E_{q}(x)=e_{\frac{1}{q}}(x)\), we obtain
Since
where we used the identity \((a;q^{-1})_{n}=(a^{-1};q)_{n}(-a)^{n}q^{-\frac{n(n-1)}{2}}\). Since \( [n]_{1/q}!=q^{\frac{n(1-n)}{2}}[n]_{q}!\), then (2.29) takes the form
Therefore,
Equating the coefficients of \( t^{n}\) in (2.30) gives (2.27) and substituting with \( x=0 \) into (2.27) yields directly (2.28). \(\square \)
Al-Salam in [3] introduced the polynomials
He also proved that
The following theorem introduces connection relations between the polynomials \(B^{(1)}_{n,\alpha }(x;q)\) and \(B^{(2)}_{n,\alpha }(x;q)\).
Theorem 2.13
For \( n \in {\mathbb {N}}_{0} \),
Proof
Since \(E_{q}(xt) e_{q}(-xt)= 1,\,\mid xt \mid <\frac{1}{1-q},\) then from (2.9), the generating function of \( B^{(1)}_{n,\alpha }(x;q) \) can be represented as
From (2.1), (2.2) and (2.33), we obtain
Therefore, equating the coefficients of the nth power of t in the series of the outside parts of (2.36) gives (2.34). The proof for \(B^{(2)}_{n,\alpha }(x;q)\) follows similarly from the generating function of \( B^{(2)}_{n,\alpha }(x;q) \) and the identity (2.32), and is omitted. \(\square \)
Theorem 2.14
Let x and \(\alpha \) be two complex numbers, with \(Re\,\alpha >-1 \), and n a positive integer. Then
Proof
We can write the generating function of the polynomials \(B^{(1)}_{n,\alpha }(x;q)\) as
On one hand, applying the Cauchy product formula in (2.38), we obtain
On the other hand, using the series representation of \(e_{q}(x)\) in (1.4) followed by the Cauchy product formula, and using (2.31) yields
Hence
equating the coefficients of \(t^{n}\) in (2.40), we get
Replacing \(x \,by\,\, \frac{-x}{2}\) in (2.41) gives
which readily completes the proof for \(B^{(1)}_{n,\alpha }(x;q)\). The proofs for \(B^{(2)}_{n,\alpha }(x;q)\) and \(B^{(3)}_{n,\alpha }(x;q)\) are similar and are omitted. \(\square \)
If we set \(x=0\) in (2.37), we obtain the following recurrence relations for \(\beta _{n,\alpha }(q)\) and \(\beta ^{(3)}_{n,\alpha }(q),\)
As a consequence of the recursive relations in (2.42), and the fact that
we can prove that
and
Theorem 2.15
For \( n\in \,{\mathbb {N}}_{0} \) and complex numbers a and x,
Proof
The proof of (2.43) follows from the generating function (2.1) since
From the q-binomial theorem (see [12, Eq.(1.3.2), p. 8]), we can prove that
Therefore,
where we used the Cauchy product formula. Equating the coefficients of \( t^{n} \) in (2.45), we obtain (2.43). The proof for \(B^{(2)}_{n,\alpha }(x;q)\) is similar and is omitted. \(\square \)
Lemma 2.16
For \(n\in {\mathbb {N}}_{0}\), \(Re \,\alpha >-1\), and \(\mid \frac{(1-q)t}{2}\mid <1\),
Proof
From Lemma 2.2, we conclude that
From the series representations of \(E_{q}(x)\) and \(g^{(1)}_{\alpha }(it;q)\) in (1.4) and (2.38), respectively we obtain
where we used the identity, see [12, Eq. (1.2.32), p. 6],
Therefore, using the identity \((a;q)_{2n}=(a;q^{2})_{n}(aq;q^{2})_{n}\) yields
Since
see [12, p. 26], then
Hence from Lemma 2.2 and (2.48), we obtain (2.46) and completes the proof. \(\square \)
Theorem 2.17
Let \( \alpha \) be a complex number such that \( Re\,\alpha >-1\). Then
where \((c_{k})_{k}\) are the coefficients defined in (2.23).
Proof
We can write Eq. (2.1) in the form
From Lemma 2.16, we obtain
where
Now, applying the Cauchy product formula in (2.49) gives
Equating the coefficients of the nth power of t in (2.52) gives
Substituting from (2.51) into (2.53), we get the result for \(B^{(1)}_{n,\alpha }(x;q)\). Similarly, we can prove the result for \( B^{(k)}_{n,\alpha }(x;q)\,(k=2,3) \). \(\square \)
Theorem 2.18
Let n be a positive integer and x be a complex number. If \(Re\,\alpha >-1 \), then
Proof
We give only the proof of (2.54) since the proof of (2.55) is similar. From (2.1),
Since
then Eq. (2.56) can be written as
Replacing x, t by \(-x, -t\), respectively in (2.39) gives
From (2.39) and (2.58), the left hand side of (2.57) can be written as
Therefore, by (2.21) and the Cauchy product formula, we get
Since the left hand side of (2.57) and (2.59) are equal, then equating the coefficients of \(t^{n}\) on the right hand sides of (2.57) and (2.59) yields (2.54) and completes the proof. \(\square \)
Proposition 2.19
If \(\alpha _{0}>-1\) satisfies the condition
then \((t/2)^{-\alpha }J_{\alpha }^{(2)}(t(1-q);q^{2})\) has no zeros in \(\mid t\mid \le 1\) for all \(\alpha \ge \alpha _{0}.\)
Proof
Set
and
Then, under hypothesis (2.60) and since \(0< q < 1\),
holds whenever \(\alpha \ge \alpha _{0}\). Hence
for \(t\in {\mathbb {R}},\, \mid t\mid \le 1\)
This proves that F(t) has no zeros on \([-1,1]\), since F(t) has only real zeros, then F(t) has no zeros in the unit disk. i.e \(\mid F(t)\mid >0,\,\,for\,\,\mid t\mid \le 1.\) \(\square \)
Corollary 2.20
There exists \(\alpha _{0}>-1\) such that \(J_{\alpha }^{(2)}(t(1-q);q^{2})\) has no zeros in the unit disk for all \(\alpha \ge \alpha _{0}\).
Proof
Since for a fixed \(q\in (0,1)\),
then there exists \(\alpha _{0}>-1 \) such that the condition (2.60) holds for all \(\alpha \ge \alpha _{0}\). Consequently from Proposition 2.19, \(J_{\alpha }^{(2)}(t(1-q);q^{2})\) has no zeros in the unit disk for all \(\alpha \ge \alpha _{0}\). \(\square \)
Theorem 2.21
For \(n\in {\mathbb {N}}\),
Proof
Taking the limit on both sides of Eq. (2.2) as \( \alpha \rightarrow \infty \) we get
From Corollary 2.20, there exists \(\alpha _{0}>-1\) such that \(g^{(2)}_{\alpha }(it;q)\) has no zeros in \(\mid t\mid \le 1\) for all \(\alpha \ge \alpha _{0}\). This means that \(\dfrac{E_{q}(xt)E_{q}(\frac{-t}{2})}{ g^{(2)}_{\alpha }(it;q)}\) is analytic in \(\mid t\mid \le 1\) for all \(\alpha \ge \alpha _{0}\). Therefore, we can interchange the limit with the summation in (2.63) when \(\mid t\mid \le 1\) to obtain
Since
then from (2.32)
Equating the coefficients of \( t^{n} \) in (2.64) gives (2.61). The proof of (2.62) follows directly from the relation (2.9) since
Hence
Therefore, computing the limit in both sides of (2.1) gives
From the q-binomial theorem (see [12, Eq.(1.3.2), p. 8]), we have
Hence
equating the coefficients of \(t^{n}\) in (2.65) yields the required result. \(\square \)
Corollary 2.22
For \( n\in {\mathbb {N}}\),
Proof
Since
then substituting with \(x=0\) into (2.62) yields (2.66). \(\square \)
Lemma 2.23
Let \(\alpha _{0}>-1\). If \(q^{3/2}(1-q)^{2}<(1-q^{2})(1-q^{2\alpha _{0}+2})\), then \((q^{\frac{1}{4}}t/2)^{-\alpha }J_{\alpha }^{(3)}(\frac{t}{2}(1-q)q^{\frac{-1}{4}};q^{2})\) has no zeros in \(\mid t\mid \le 1\) for all \(\alpha \ge \alpha _{0}.\)
Proof
The proof is similar to the proof of Proposition 2.19 and is omitted. \(\square \)
Theorem 2.24
For \( n\in {\mathbb {N}}\),
Proof
Taking the limit as \(\alpha \rightarrow \infty \) on both sides of (2.3), we obtain
We can choose \(\alpha _{0}>-1\) such that
for all \(\alpha \ge \alpha _{0}\). Hence from Lemma 2.23, the function \(g^{(3)}_{\alpha }(it;q)\) does not vanish on the unit disk, and the left hand side of (2.69) is analytic for \(\mid t\mid \le 1\). Therefore, we can interchange the limit as \(\alpha \rightarrow \infty \) with the summation in (2.69) to obtain
Since
Hence
But
Therefore,
Substituting from (2.71) into (2.70) and equating the coefficients of \(t^{n} \) yields (2.67). The proof of (2.68) follows directly by setting \(x=0\) in (2.67). \(\square \)
Theorem 2.25
Let \(\alpha \) be a complex number such that \( Re\,\alpha >-1\). Then for \(n\in {\mathbb {N}},\,n\,\ge \,2\),
Proof
We give in detail the proof of \(\beta _{n,\alpha }(q)\) in (2.72). The proof for \(\beta _{n,\alpha }^{(3)}(q)\) is similar. Since
then
Consequently, from the series representation of \(e_{q}(t)\) in (1.4), we get
Since
then substituting from (2.75) into (2.74) and using (2.73), we obtain
Therefore, by the Cauchy product formula
Equating the coefficient of \( t^{n}\) in (2.76), we get the required result and the theorem follows. \(\square \)
The following theorem gives a recursive relations between the polynomials \(B^{(k)}_{n,\alpha }(x;q)\) and \(B^{(k)}_{n,\alpha +1}(x;q)\,(k=2,3)\).
Theorem 2.26
If \( Re\,\alpha > -1 \) , \( x\in {\mathbb {C}},\) and \( k \in \,{\mathbb {N}}\), then
where
and \((j^{(r)}_{m,\alpha })_{m=1}^{\infty }\,\,(r=2,3)\) are the positive zero of \( J^{(r)}_{\alpha }( \cdot ;q^{2}).\)
Proof
We start with the proof of (2.77). From [6, 13], we have the identity
where
Replacing t by \( it(1-q)\) and q by \( q^{2}\) in (2.79), we obtain
Multiplying (2.80) by \( E_{q}(xt) E_{q}(\frac{-t}{2})\) to obtain
Substituting from (2.8) into (2.81), we get
Consequently,
Hence
Equating the coefficients of \( t^{n}\) in (2.83), we get (2.77). The proof of (2.78) follows from the identity (see [1, Eq. (4.3), p. 1201]),
where
and by using the same technique. \(\square \)
3 Asymptotic Relations for the Generalized q-Bernoulli Numbers
In this section, we derive asymptotic relations for the generalized q-Bernoulli numbers defined in (2.10).
Theorem 3.1
Let n be a non negative integer and \(\alpha \) be a complex number such that \(Re\,\alpha >-1\). Then for \(n\in {\mathbb {N}},\)
where \({\mathcal {J}}_{\alpha }^{(2)} (z;q)\) is defined in (1.7).
Proof
Since
then
Now, we integrate \(f(z):=\dfrac{G(z)}{z^{n+1}},\,G(z)=\dfrac{E_{q}(\frac{-z}{2})}{g^{(2)}_{\alpha }(iz;q)}\) on the contour \(\Gamma _{m},\) where \(\Gamma _{m}\) is a circle of radius \(R_{m},\) \(\mid z_{m}\mid<\,R_{m}<\mid z_{m+1}\mid \). From the Cauchy Residue Theorem, see [2],
where \(\{z_{k}\}\) are the poles of f that lie inside \(\Gamma _{m}\). The function f(z) has a pole at \(z=0\) of order \(n+1\) and simple poles at \(\pm z_{k}\) where \(z_{k}= \dfrac{ij^{(2)}_{k,\alpha }}{1-q},\,k\in {\mathbb {N}}\). Consequently,
Since
and
Then Eq. (3.2) can be written as
substituting into (3.3) with \(-i=e^{-\frac{i\pi }{2}}\) gives
Now, we show that the integral \(I_{m}\rightarrow 0\) as \(m\rightarrow \infty \). Bergweiller and Hayman [7] introduced the asymptotic relation for \(E_{q}(z)\),
In [4], Annaby and Mansour proved that for \(r=\mid z\mid \,\rightarrow \,\infty \)
Hayman in [15] introduced the higher order asymptotics of \(J_{\nu }^{(2)}(z;q)\). Then, Annaby and Mansour, see [4], pointed out that the first order asymptotics of the zeros of \(J_{\nu }^{(2)}(z;q^{2})\) is given by
Hence if \((z_{m})_{m}\) are the positive zeros of \(g^{(2)}_{\alpha }(iz;q)\), then
Let \( 0<\epsilon <(q^{-1}-1)\). There exists \(M_{0}\in {\mathbb {N}}\) such that if \(m\in {\mathbb {N}},\,m\ge M_{0}\), then
Hence \(z_{m}<qz_{m+1}\) for all \(m\ge M_{0}\). We can choose \(R_{m},\,\delta :=q^{-1}\displaystyle \sup _{m\ge M_{0}}\frac{z_{m}}{z_{m+1}}\) such that \((z_{m}<\delta R_{m}<q z_{m+1} <R_{m})\). Indeed,
But \(q z_{m+1} <R_{m}\) leads to \(\delta >\frac{z_{m}}{R_{m}}\) and so \( z_{m} <\delta R_{m}\). Now,
Also \(\delta =q^{-1}\displaystyle \sup _{ m\ge M_{0}}\frac{z_{m}}{z_{m+1}}\le q(1+\epsilon )<1\). Hence \(1>\delta \ge q\) and so by
the annulus \(\delta R_{m}<|z| <R_{m}\) has no zeros of the function \( g^{(2)}_{\alpha }(iz;q)\). Hence, from the minimum modulus principle we have
Therefore, from (3.6), we conclude that
where
Now, using the ML-inequality (see [2]) to obtain
From (3.4) and (3.5), we have \(\displaystyle \lim _{m\rightarrow \infty }R_{m}=\infty \). Also, since \(0<q<1\) and \(1>\delta \ge q\) then
Hence \(\displaystyle \lim _{m\rightarrow \infty }I_{m}=0.\) Consequently,
Therefore,
which completes the proof of the theorem. \(\square \)
Remark 3.2
If we substitute with \(\alpha =\frac{1}{2}\) in the second equation in (3.1), then \((z_{k})_{k}\) will be the positive zeros of \(Sin_{q}(z)\) and consequently, the series in the left hand side vanishes which coincide with the known result that the odd Bernoulli numbers vanish \((\beta _{2n+1}(q)=0,\,n\ge \, 1)\) (see [19]). Similarly, if we set \(\alpha =-\frac{1}{2}\) in the first equation in (3.1), the series in the left hand side vanishes and this coincide with the fact that the even Euler\(^{,}\)s numbers are zero \((E_{2n}(q)=0,\,n\ge \,1)\) (see [19]).
Corollary 3.3
The asymptotic relations of the generalized q-Bernoulli numbers \((\beta _{n,\alpha }(q))_{n},\)
where \({\mathcal {J}}_{\alpha }^{(2)} (z;q)\) is defined in (1.7).
Proof
The proof follows directly from Theorem 3.1. \(\square \)
4 Applications of the Generalized q-Bernoulli Polynomials
In this section, we introduce connection relations between the generalized q-Bernoulli polynomials \( B^{(k)}_{n,\alpha }(x;q)\,(k=1,2,3)\) and the q-Laguerre and the little q-Legendre polynomials.
The q-Laguerre polynomials \( L_{n}^{\alpha }(x;q)\) of degree n are defined by
The Rodrigues formula is given by
and the orthogonality relation is
\(\alpha >-1\), where \(\delta _{mn}\) is the Kronecker delta function, see [17, 21]. The q-Laguerre polynomials \( L_{n}^{\alpha }(x;q)\) satisfy three term recurrence relation
where
In the following, let \(\alpha >-1\) and \({\mathbb {P}}_{n}=\{p(x):\text{ deg }\, p(x)\le n\}\) with the inner product
where \(p(x),\,g(x)\in {\mathbb {P}}_{n}\).
Theorem 4.1
Let p(x) \(\in {\mathbb {P}}_{n}.\) Then p(x) can be expanded as
where
Proof
Since
in order to calculate the constant \(C_{m}\), we use (4.3) to obtain
Then
Therefore,
Using (4.2) with n replaced by m, we obtain
and the theorem follows. \(\square \)
The following Lemma, see [18], is essential in the proof of Theorem 4.3.
Lemma 4.2
Let the functions f and g be defined and continuous on \([0,\infty ]\). Assume that the improper Riemann integrals of the functions f(x)g(x) and f(x/q)g(x) exist on \([0,\infty ]\). Then
Theorem 4.3
If \(n\in {\mathbb {N}}\) and \(x\in {\mathbb {C}},\) then
where
Proof
We prove the identity for \(B^{(1)}_{n,\alpha }(x;q)\) and the proofs for \(B^{(k)}_{n,\alpha }(x;q)\,(k=2,3)\) are similar. Substitute with \(p(x)=B^{(1)}_{n,\alpha }(x;q)\) in (4.4). This gives
Since \(\{L_{m}^{\alpha }(x;q)\}_{n\in {\mathbb {N}}}\) is an orthogonal polynomials sequence then \( C_{m}=0\) for \( m>n\), and
Now, we calculate \(C_{m}\). Using (2.12) in (4.5) gives
Since
then
From (4.2), we get
then applying the q-integration by part introduced in Lemma 4.2m times, we obtain
From [16, Eq. (5.4), p. 465],
Then
Therefore,
Since
then substituting from (4.7) into (4.6), we get
Using the relation (2.47), we obtain
and this completes the proof of the theorem. \(\square \)
The little q-Legendre polynomials \(( P_{n}(x\mid q))_{n}\) are defined by
They satisfy the Rodrigues formula
and the orthogonality relation
see [21]. Let \({\mathbb {P}}_{n}=\{g(x):\text{ deg }\, g(x)\le n\}\) with the inner product
where \(p(x),\,g(x)\in {\mathbb {P}}_{n}\).
Theorem 4.4
Let g(x) \(\in {\mathbb {P}}_{n}\). Then g(x) can be represented by
where
Proof
Since
then by the orthogonality relation (4.9), we obtain
By using (4.8), we get
which readily gives the result. \(\square \)
Theorem 4.5
For \(n\in {\mathbb {N}}\) and \(x\in {\mathbb {C}}\),
where
Proof
Substitute with \(g(x)=B^{(1)}_{n,\alpha }(x;q)\) in (4.10), we obtain
Since the polynomials \(\{P_{k}(x\mid q)\}\) are orthogonal, then \( C_{k}=0\) for \(k>n\), and
Set
From (4.11),
since
Hence, by the Rodrigues formula in (4.8), we obtain
Using the \(q^{-1}\)-integration by parts
where f and g are continuous functions at zero, see [5]. This gives
The first term on the right hand side of (4.15) vanishes because
and
Therefore,
Now, applying (4.14) \(k-1\) times on the right hand side of (4.16), and using that \(D_{q^{-1}}^{m}(x^{k}(qx;q)_{k}= 0\) at \(x=0,\, x=\frac{1}{q}\) \((m=0,1,\ldots ,k-1)\) yields
Since
see [5, Eq. (1.58), p. 22], then
Substituting from (4.17) into (4.13) yields
where we used the identity in (2.47). Therefore, from (4.18) and (4.12), we get the required result for \(B^{(1)}_{n,\alpha }(x;q)\). Similarly, we can prove the result for \(B^{(k)}_{n,\alpha }(x;q)\,(k=2,3)\). \(\square \)
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Eweis, S.Z.H., Mansour, Z.S.I. Generalized q-Bernoulli Polynomials Generated by Jackson q-Bessel Functions. Results Math 77, 132 (2022). https://doi.org/10.1007/s00025-022-01656-x
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DOI: https://doi.org/10.1007/s00025-022-01656-x