Abstract
The purpose of this work is to demonstrate several interesting contiguous function relations and q-differential formulas for basic Horn hypergeometric functions \(\mathbf{H}_{3}\) and \(\mathbf{H}_{4}\). Some properties of our main results are also constructed.
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1 Introduction
Quantum calculus (or, q-calculus) is the study of calculus without limits. Many extensions of q-calculus have been developed and applied as one of the most active areas of research in mathematics and physics. These new extensions have proved to be very useful in various fields such as physics, engineering, statistics, actuarial sciences, economics, survival analysis, life checking out and telecommunications, and many others (see for example [17, 23, 24, 31, 33]). The applications have largely stimulated our present study. One of the most important branches of q-calculus is q-special functions. Jackson [19, 20], Andrews [4, 5], Gupta [16], Agarwal [2, 3], Ismail and Libis [18], Jain [21, 22], Jain and Vertna [23], Mishra [24], Sahai and Verma [25–27], Srivastava [30], Srivastava and Jain [31], Swarttouw [33], Verma and Sahai [34] introduced and discussed some interesting properties for various families of the basic Appell series, basic hypergeometric series, and q-Lauricella series by applying certain operators of q-calculus and its applications. Acikgoz et al. [1] and Araci et al. [6, 7] introduced a class of q-Euler, q-Frobenius–Euler, and q-Bernoulli polynomials based on q-exponential functions. Duran et al. [13, 14] introduced q-Bernoulli, q-Euler, and q-Genocchi polynomials and obtained the q-analogues of familiar earlier formulas and identities. In [8], Bagdasaryan et al. constructed Apostol q-Bernoulli, Apostol q-Genocchi, and Apostol q-Euler polynomials. Bansal and Choi [9], Bansal and Kumar [10], and Bansal et al. [11, 12] introduced and investigated the Pathway fractional integral formulas, fractional integral operators and integral transform of incomplete H-functions, incomplete ℵ-functions, incomplete I-functions, and S-generalized Gauss hypergeometric function. In [28, 29], Shehata investigated and discussed the generating functions for \((p,q)\)-Bessel and \((p,q)\)-Humbert functions.
Throughout this paper, we observe the following notations \(0<|q|<1\), \(q\in \mathbb{C}-\{1\}\). Let \(\mathbb{N}\) and \(\mathbb{C}\) be the sets of natural numbers and complex numbers, respectively.
The q-number (basic or quantum number) \([\beta ]_{q}\) is defined by [5, 15, 17]
The q-number and q-factorial are given by
and
where \((\beta ;q)_{n}\) is the q-shifted factorial (q-Pochhammer symbol) which is definoted as follows: for \(n\in \mathbb{N}\), \(\beta \in \mathbb{C}\setminus \{1, q^{-1}, q^{-2},\ldots ,q^{1-n}\}\), \(0<|q|<1\), \(q\in \mathbb{C}-\{1\}\);
Note that taking limit as tends to (1.1) in the above relations gives the shifted factorial \((\beta )_{n}\) (see [32])
We recall some notations and defifinitions from q-calculus for \(\beta \in \mathbb{C}\), \(n\in \mathbb{N}\), \(0<|q|<1\), \(q\in \mathbb{C}-\{1\}\), which are essential in the sequel (see [15]):
and
Definition 1.1
Based on q-Pochhammer’ symbol (1.4), we define the basic Horn hypergeometric functions \(\mathbf{H}_{3}\) and \(\mathbf{H}_{4}\) as follows:
and
Remark 1.1
If \(q\rightarrow 1\), the basic Horn hypergeometric functions reduce to the Horn hypergeometric functions defined in [[32], p. 56, Eq. (27), p. 57, Eq. (28)].
To simplify the notation, we write \(\mathbf{H}_{3}\) for the series \(\mathbf{H}_{3}(a,b;c;q,x,y)\), \(\mathbf{H}_{3}(aq^{\pm 1})\) for the series \(\mathbf{H}_{3}(aq^{\pm 1},b;c;q,x,y)\), \(\mathbf{H}_{4}\) for the series \(\mathbf{H}_{4}\), … , and \(\mathbf{H}_{4}(cq^{\pm 1})\) stands for the series \(\mathbf{H}_{4}(a;b,cq^{\pm 1};q,x,y)\).
For a wide variety of other investigations involving basic Horn hypergeometric functions, see, for instance, [2, 3, 5, 30]. Motivated by the previous works in q-analysis (see [25, 26, 34]), in this paper we introduce a class of new extended forms of the basic Horn hypergeometric function. Our study can be detailed as follows: In Sect. 2, we introduce and study some contiguous functions relations and q-differential formulas for our considered basic Horn hypergeometric functions \(\mathbf{H}_{3}\) by permuting parameters. We discuss some family relations between basic Horn hypergeometric functions \(\mathbf{H}_{4}\) in Sect. 3. Finally, we discuss our main results and related results involving contiguous relations for \(\mathbf{H}_{3}\) and \(\mathbf{H}_{4}\) in Sect. 4.
2 Contiguous functions for \(\mathbf{H}_{3}(a,b;c;q,x,y)\)
Here we derive several properties as well as the contiguous function relations for \(\mathbf{H}_{3}\) with \(c\neq 1, q^{-1}, q^{-2},\ldots \) .
Theorem 2.1
For \(c\neq 1\), the contiguous relations of \(\mathbf{H}_{3}\) hold true for the numerator parameter a:
and
Proof
Replacing a by aq in (1.8), we get
The relations
and
imply
By using the relation \(1-q^{2m+n}=1-q^{n}+q^{n}(1-q^{m})+q^{m+n}(1-q^{m})\), we get (2.2). Performing the replacement \(a\rightarrow aq^{-1}\) in the contiguous relations (2.1) and (2.2), we obtain (2.3) and (2.4). □
Theorem 2.2
For \(c\neq 1\), \(\mathbf{H}_{3}\) satisfies the derivative equations
and
Proof
From (1.8), we consider the operators \(\theta _{x,q}=x\frac{\partial }{\partial x}=xD_{x,q}\) and \(\theta _{y,q}=y\frac{\partial }{\partial y}=yD_{y,q}\) to get
and
□
Theorem 2.3
\(\mathbf{H}_{3}\) satisfies the difference equations
and
Proof
With the help of the above differential operators and using (2.5) and (2.6) for \(\mathbf{H}_{3}\), we get the results (2.7)–(2.10). □
Theorem 2.4
For \(c\neq 1\), the contiguous function relations of \(\mathbf{H}_{3}\) with the numerator parameter b give
and
Proof
If we replace b by bq in (1.8), we get
Using the relations
and
we have
Replacing \(b\rightarrow bq^{-1}\) in relation (2.11), we obtain (2.12). □
Theorem 2.5
The difference equations hold true for \(\mathbf{H}_{3}\):
and
Proof
From (2.5) and (2.6), we get (2.13) and (2.14). □
Theorem 2.6
For \(c\neq 1\), \(b\neq q\), the formulas hold true for \(\mathbf{H}_{3}\):
and
Proof
Using (1.8), (1.5), (1.6), (1.7) and the relation \((bq^{-1};q)_{n}=(1-bq^{-1})(b;q)_{n-1}\) implies
Using the relation \(a(1-q^{2m+n})-abq^{n-1}(1-q^{2m})-bq^{-1}(1-q^{n})=a(1-q^{m})+aq^{m}(1-q^{m})+aq^{2m}(1-q^{n}) -abq^{n-1}(1-q^{m})-abq^{m+n-1}(1-q^{m})-bq^{-1}(1-q^{n})\), we obtain (2.16). □
Theorem 2.7
Each of the following properties for \(\mathbf{H}_{3}\) in (1.8) holds true:
Proof
From (1.8), we have
By using the equation
and
we get
□
Theorem 2.8
The contiguous function relations of \(\mathbf{H}_{3}\) with denominator parameter c are valid:
and
Proof
By the definition of basic Horn function, we get
Using
and
we can rewrite the above equation as follows:
which is the desired result. The proof of Eq. (2.19) can run parallel to Eq. (2.18), so details are omitted here.
If we replace \(c\rightarrow cq\) in relation (2.18), we obtain
The proof of Eq. (2.21) would run parallel to Eq. (2.20), so we may skip the involved details. □
Theorem 2.9
The derivative formulas for \(\mathbf{H}_{3}\) are satisfied:
and
Proof
Using the definition \(\mathbf{H}_{3}\) in (1.8) with the relation \(\frac{1}{(cq^{-1};q)_{m+n}}=\frac{1}{(c;q)_{m+n}} {[} \frac{c}{c-q}q^{m+n}-\frac{q}{c-q} {]}\), we get
Replacing \(c=cq\) in (2.22) implies the contiguous relation
By using Eqs. (2.5) and (2.6), we obtain the required results (2.23) and (2.24). □
Theorem 2.10
For \(c\neq 1\) and \(cq\neq 1\), the following formulas are valid:
and
Proof
Replacing a and c by aq and cq in (1.8), we get
By using the relations
The proof of Eqs. (2.27)–(2.29) would run parallel to Eq. (2.26) by using the above relations, so we omit the involved details. □
Theorem 2.11
For \(\mathbf{H}_{3}\) defined by (1.8), each of the formulas holds:
and
for \(c\neq 1\) and \(cq\neq 1\).
Proof
From (1.8), we have
Hence, we obtain (2.30), one can derive the result (2.31) by a similar way. □
3 Contiguous functions for \(\mathbf{H}_{4}\)
Relying on a similar procedure as the one used in the previous section, we obtain the following list of results for basic Horn function \(\mathbf{H}_{4}(a,b;c,d;q,x,y)\) with \(c,d\neq 1, q^{-1}, q^{-2},\ldots \) .
Theorem 3.1
For \(c\neq 1\) and \(d\neq 1\), the contiguous function relations of \(\mathbf{H}_{4}\) with numerator parameter a hold true:
and
Proof
In (1.9), replacing a by aq, we get
Using the relations
and
we get
The proof of Eq. (3.2) would run parallel to Eq. (3.2), so details are omitted here. Performing the replacement \(a\rightarrow aq^{-1}\) in the contiguous relations (3.1) and (3.2), we get (3.3) and (3.4). □
Theorem 3.2
The q-derivatives of \(\mathbf{H}_{4}\) defined in (1.9) are valid:
and
Proof
In (1.9), we apply the operators \(\frac{\partial }{\partial x}=D_{x,q}\) and \(\frac{\partial }{\partial y}=D_{y,q}\) to get
and
Iterating this technique n times on \(\mathbf{H}_{4}\), we obtain (3.5) and (3.6). □
Theorem 3.3
For \(\mathbf{H}_{4}\) defined in (1.9), we have
and
Proof
By using these q-derivatives of \(\mathbf{H}_{4}\) in (3.5) and (3.6), we get the recursion formulas (3.7) and (3.8).
Using (3.5), (3.7), and (3.8) for \(\mathbf{H}_{4}\), we obtain (3.9) and (3.10). □
Theorem 3.4
For \(d\neq 1\), the contiguous function relations of \(\mathbf{H}_{4}\) with the numerator parameter b hold true:
and
Proof
In (1.9), we replace b by bq to obtain
By means of the relations
and
we have
Replacing \(b\rightarrow bq^{-1}\) in relation (3.12), we get
□
Theorem 3.5
The q-differential formulas are valid:
and
Proof
From (3.5) and (3.6), we obtain (3.13) and (3.14). □
Theorem 3.6
For \(c\neq 1\), \(d\neq 1\), and \(b\neq q\), the contiguous function relations hold true for the numerator parameters a and b of \(\mathbf{H}_{4}\):
and
Proof
Using the definition of \(\mathbf{H}_{4}\) with the relation \((bq^{-1};q)_{n}=(1-bq^{-1})(b;q)_{n-1}\), we get
Proceeding as above, one can get the proof of Eq. (3.16) that would run parallel to Eq. (3.15), so details are omitted here. □
Theorem 3.7
For \(c\neq 1\), the recursion formulas hold true for the numerator parameter b of \(\mathbf{H}_{4}\):
Proof
From (1.9) we have
By using the equation
and
we get
□
Theorem 3.8
The contiguous function relations of \(\mathbf{H}_{4}\) with denominator parameters c and d hold true:
and
Proof
By the definition of \(\mathbf{H}_{4}\), we have
Using
and
we get
which is the desired result. The recursion formula (3.19) can be proved in a similar manner. □
Theorem 3.9
The contiguous function relations hold true for the denominator parameter c of \(\mathbf{H}_{4}\):
Proof
Replacing \(c\rightarrow cq\) in (3.18) and \(d\rightarrow dq\) in (3.19), we obtain (3.20). □
Theorem 3.10
The following results of \(\mathbf{H}_{4}\) hold well:
and
Proof
We replace \(\frac{1}{(cq^{-1};q)_{m}}=\frac{1}{(c;q)_{m}} {[}\frac{c}{c-q}q^{m}- \frac{q}{c-q} {]}\) in relation (1.9) to get
Replacing \(c=cq\) in (3.21), we get
The results (3.25), (3.26), (3.22), and (3.24) are along the same lines as those of Eqs. (2.22)–(2.25). □
Theorem 3.11
For \(c, d, cq \neq 1\), the results of \(\mathbf{H}_{4}\) hold:
and
Proof
In (1.9), replacing a and c by aq and cq, we have
By using the relation
The proof of Eq. (3.27) is similar as that of Eq. (3.28). □
Theorem 3.12
For \(c, d, cq \neq 1\), the following contiguous function relations hold:
and
Proof
By virtue of our calculation in Theorem 3.11, and with the help of the relations
and
To simplify the above relationships, we get (3.29) and (3.30). □
Theorem 3.13
For \(c, d, dq \neq 1\), the contiguous relations hold true for the parameters of \(\mathbf{H}_{4}\):
and
Proof
Using the relations
and
Simplifying the above relations, we obtain (3.31)–(3.34). □
Theorem 3.14
The formulas hold true of \(\mathbf{H}_{4}\):
and
Proof
By using the definition of \(\mathbf{H}_{4}\), we get
We prove relation (3.36) in a similar way as relation (3.35). □
4 Conclusion
We conclude by the remark that the results established in this paper are general forms and one can deduce several contiguous function relations and q-differential relations of basic Horns hypergeometric functions \(\mathbf{H}_{3}\) and \(\mathbf{H}_{4}\) as different cases of our main findings. Also, other types of these extensions are recommended for a parallel study of this work. More work will be carried out in the coming future results in other fields of interest for fractional quantum calculus.
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References
Acikgoz, M., Araci, S., Duran, U.: New extensions of some known special polynomials under the theory of multiple q-calculus. Turk. J. Anal. Number Theory 3, 128–139 (2015)
Agarwal, R.P.: Some relations between basic hypergeometric functions of two variables. Rend. Circ. Mat. Palermo (2) 3, 1–7 (1954)
Agarwal, R.P.: Contiguous relations for bilateral basic hypergeometric series. Int. J. Math. Sci. 3(2), 375–388 (2004)
Andrews, G.E.: On q-difference equations for certain well-poised basic hypergeometric series. Quart. J. Math. 19(2), 433–447 (1968)
Andrews, G.E.: Problems and Prospects for Basic Hypergeometric Functions. In: Askey, R.A. (ed.) Theory and Applications of Special Functions. Academic Press, San Diego (1975)
Araci, S., Acikgoz, M., Park, K.H., Jolany, H.: On the unification of two families of multiple twisted type polynomials by using q-adic q-integral at \(q=-1\). Bull. Malays. Math. Sci. Soc. 37, 17–41 (2014)
Araci, S., Duran, U., Acikgoz, M.: Symmetric identities involving q-Frobenius-Euler polynomials under Sym (4). Turk. J. Anal. Number Theory 3, 90–93 (2015)
Bagdasaryan, A., Araci, S., Acikgoz, M., He, Y.: Some new identities on the Apostol–Bernoulli polynomials of higher order derived from Bernoulli basis. J. Nonlinear Sci. Appl. 9, 2697–2704 (2016)
Bansal, M.K., Choi, J.: A note on pathway fractional integral formulas associated with the incomplete H-functions. Int. J. Appl. Comput. Math. 5(5), 133 (2019)
Bansal, M.K., Kumar, D.: On the integral operators pertaining to a family of incomplete I-functions. AIMS Math. 5(2), 1247–1259 (2020)
Bansal, M.K., Kumar, D., Jain, R.: A study of Marichev–Saigo–Maeda fractional integral operators associated with S-generalized Gauss hypergeometric function. Kyungpook Math. J. 59(3), 433–443 (2019)
Bansal, M.K., Kumar, D., Nisar, K.S., Singh, J.: Certain fractional calculus and integral transform results of incomplete ℵ-functions with applications. Math. Methods Appl. Sci. 43(8), 5602–5614 (2020)
Duran, U., Acikgoz, M., Araci, S.: On a class of multiple q-Bernoulli, multiple q-Euler and multiple q-Genocchi polynomials of order α. Adv. Appl. Math. Sci. 15(8), 229–254 (2016)
Duran, U., Acikgoz, M., Esi, A., Araci, S.: Some new symmetric identities involving q-Genocchi polynomials under \(S_{4}\). J. Math. Anal. 6(5), 22–31 (2015)
Gasper, G., Rahman, M.: Basic Hypergeometric Series, 2nd edn. Encyclopedia of Mathematics and Its Applications, vol. 96. Cambridge University Press, Cambridge (2004)
Gupta, D.: Contiguous relations, basic hypergeometric functions, and orthogonal polynomials. III. Associated continuous dual q-Hahn polynomials. J. Comput. Appl. Math. 68(1–2), 115–149 (1996)
Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and Its Applications, vol. 98. Cambridge University Press, Cambridge (2009)
Ismail, M.E.H., Libis, C.A.: Contiguous relations, basic hypergeometric functions, and orthogonal polynomials I. J. Math. Anal. Appl. 141(2), 349–372 (1989)
Jackson, F.H.: On basic double hypergeometric functions. Quart. J. Math. 13, 69–82 (1942)
Jackson, F.H.: On some formulae in partition theory, and bilateral basic hypergeometric series. J. Lond. Math. Soc. 24, 233–237 (1949)
Jain, V.K.: Some transformations of basic hypergeometric series and their applications. Proc. Am. Math. Soc. 78, 375–384 (1980)
Jain, V.K.: Some transformations of basic hypergeometric functions. Part II. SIAM J. Math. Anal. 12, 957–961 (1981)
Jain, V.K., Vertna, A.: Transformations between basic hypergeometric series on different bases and identities of Rogers–Ramanujan type. J. Math. Anal. Appl. 76, 230–269 (1980)
Mishra, B.P.: On certain transformation formulae involving basic hypergeometric functions. J. Ramanujan Soc. Math. Math. Sci. 2(2), 9–16 (2014)
Sahai, V., Verma, A.: nth-order q-derivatives of multivariable q-hypergeometric series with respect to parameters. Asian-Eur. J. Math. 7(2), 1450019 (2014)
Sahai, V., Verma, A.: nth-order q-derivatives of Srivastava’s general triple q-hypergeometric series with respect to parameters. Kyungpook Math. J. 56, 911–925 (2016)
Sahai, V., Verma, A.: Recursion formulas for q-hypergeometric and q-Appell series. Commun. Korean Math. Soc. 33(1), 207–236 (2018)
Shehata, A.: On the \((p,q)\)-Humbert functions from the view point of the generating functions method. J. Funct. Spaces 2020, Article ID 4794571 (2020)
Shehata, A.: On the \((p,q)\)-Bessel functions from the view point of the generating functions method. J. Interdiscip. Math., 1–14 (2020)
Srivastava, B.: A note on certain bibasic q-Appell and Lauricella series. Glas. Mat. 30(1), 29–36 (1995)
Srivastava, H.M., Jain, V.K.: q-series identities and reducibility of basic double hypergeometric functions. Can. J. Math. 38(1), 215–231 (1986)
Srivastava, H.M., Manocha, H.L.: A Treatise on Generating Functions. Halsted, New York (1984)
Swarttouw, R.F.: The contiguous function relations for basic hypergeometric function. J. Math. Anal. Appl. 149(1), 151–159 (1990)
Verma, A., Sahai, V.: Some recursion formulas for q-Lauricella series. Afr. Math. 30(9), 1–44 (2019)
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Shehata, A. On basic Horn hypergeometric functions \(\mathbf{H}_{3}\) and \(\mathbf{H}_{4}\). Adv Differ Equ 2020, 595 (2020). https://doi.org/10.1186/s13662-020-03056-3
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DOI: https://doi.org/10.1186/s13662-020-03056-3