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Contiguous relations for the Fox–Wright function
Accepted Manuscript Contiguous relations for the Fox–Wright function
Xiaoxia Wang
PII: DOI: Reference:
S0022-247X(19)30159-3 https://doi.org/10.1016/j.jmaa.2019.02.035 YJMAA 22968
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
6 April 2018
Please cite this article in press as: X. Wang, Contiguous relations for the Fox–Wright function, J. Math. Anal. Appl. (2019), https://doi.org/10.1016/j.jmaa.2019.02.035
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CONTIGUOUS RELATIONS FOR THE FOX–WRIGHT FUNCTION XIAOXIA WANG† Department of Mathematics, Shanghai University, Shanghai 200444, P. R. China
Contiguous relations are a powerful tool for dealing with hypergeometric functions. In this paper, four different types of contiguous relations for the Fox–Wright function are given. We demonstrate the usefulness of our formulas by showing how they can be used to recover the famous Hagen–Rothe convolution. Keywords: Fox–Wright function; Gamma function; contiguous relation; Hagen–Rothe convolution. 2010 Mathematics Subject Classification: Primary 33C20; Secondary 33C99.
1. Introduction The Pochhammer symbol (or the rising shifted factorial, since (1)n = n!) defined, for any complex number α, by ⎧ Γ(α + n) ⎨α(α + 1) · · · (α + n − 1), n ∈ N = 1, 2, 3, · · ·, = (α)n := ⎩1, Γ(α) n = 0, where the Γ–function [1], with the complex numbers x = 0, −1, −2, · · · , being defined as ∞ Γ(x) = ux−1 e−u du, R(x) > 0. 0
The following relation about the Γ–function Γ(x + 1) = xΓ(x),
(1)
will be used frequently in this paper. The Fox–Wright function [11, 17, 21, 22], which is regarded as a generalization of hypergeometric function, is defined as follows: ∞ Πpi=1 Γ(αi + Ai k) z k (α1 , A1 ), · · · , (αp , Ap ) , (2) Ψ z := p q (β1 , B1 ), · · · , (βq , Bq ) Πqj=1 Γ(βj + Bj k) k! k=0
where the coefficients {Ai }i≥1 and {Bj }j≥1 are positive real numbers with 1+
q j=1
Bj −
p i=1
E-mail addresses: [email protected] (X. Wang).
Ai ≥ 0.
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X. Wang
Recently, many summations for the Fox–Wright function are established in [7, 19]. The importance of the Fox–Wright function can be found in [8, 9, 13, 14, 15]. Recently, Opps et al. [16] established the recursion formulas of Appell’s function F2 . Brychkov and Saad [2, 3, 4, 5] researched Appell’s four functions F1 , F2 , F3 and F4 including the recursion formulas. Many contiguous relations of 3 F2 -series are given by Whipple [20] and Krattenthaler and Rivoal [12]. The author with the partner established many contiguous relations of 3 F2 series by Abel’s method [6] and recursion formulas of Appell’s functions [18]. In this paper, we present four different types of contiguous relations for the Fox–Wright function.
2. Contiguous relations with differing numerator parameters of the Fox–Wright function 2.1. In this section, we first establish the contiguous relation about the numerator α1 , then we get the contiguous formulas about the numerator parameter α1 with two expressions. The contiguous relations about the other numerator parameters can be established by the similarly method. Theorem 1. The following contiguous relations of the Fox–Wright function hold true. (α1 + k, A1 ), · · · , (αp , Ap ) (α1 , A1 ), · · · , (αp , Ap ) z = (α1 )k p Ψq z p Ψq (β1 , B1 ), · · · , (βq , Bq ) (β1 , B1 ), · · · , (βq , Bq ) +A1 z p Ψq
=
i=1
(α1 )i
p Ψq
(α1 + i − 1 + A1 , A1 ), (α2 + A2 , A2 ) · · · , (αp + Ap , Ap ) z ; (3) (β1 + B1 , B1 ), · · · , (βq + Bq , Bq )
(α1 + k, A1 ), · · · , (αp , Ap ) z (β1 , B1 ), · · · , (βq , Bq )
k i=0
k (α1 )k
(A1 z)i
(α1 + iA1 )ji (α1 + (i − 1)A1 + 1 + ji )ji−1 · · ·
l+j1 +···+ji =k−i;l,j1 ,··· ,ji ≥0
×(α1 + A1 + i − 1 + ji + ji−1 + · · · + j2 )j1 (α1 + i + ji + ji−1 + · · · + j1 )l (α1 + iA1 , A1 ), · · · , (αp + iAp , Ap ) ×p Ψq z . (β1 + iB1 , B1 ), · · · , (βq + iBq , Bq ) Proof. From the definition of the Fox–Wright function (2), we have ∞ Γ(α1 + 1 + A1 k)Πpi=2 Γ(αi + Ai k) z k (α1 + 1, A1 ), · · · , (αp , Ap ) . Ψ z = p q (β1 , B1 ), · · · , (βq , Bq ) Πqj=1 Γ(βj + Bj k) k!
(4)
(5)
k=0
Recalling the relation of the Γ–function (1), we get Γ(α1 + 1 + A1 k) = Γ(α1 + A1 k)(α1 + A1 k) = α1 Γ(α1 + A1 k) + A1 kΓ(α1 + A1 k). Then the left-hand side of (5) can be expressed as the following contiguous relation (α1 + 1, A1 ), · · · , (αp , Ap ) (α1 , A1 ), · · · , (αp , Ap ) z = α1 p Ψq z p Ψq (β1 , B1 ), · · · , (βq , Bq ) (β1 , B1 ), · · · , (βq , Bq ) (α1 + A1 , A1 ), · · · , (αp + Ap , Ap ) +A1 z p Ψq z . (β1 + B1 , B1 ), · · · , (βq + Bq , Bq )
(6)
Contiguous relations for the Fox–Wright function
Performing the replacement α1 → α1 + 1 in the above contiguous relation, we have (α1 + 2, A1 ), · · · , (αp , Ap ) (α1 + 1, A1 ), · · · , (αp , Ap ) Ψ + 1) Ψ z = (α z p q 1 p q (β1 , B1 ), · · · , (βq , Bq ) (β1 , B1 ), · · · , (βq , Bq ) (α1 + 1 + A1 , A1 ), · · · , (αp + Ap , Ap ) +A1 z p Ψq z . (β1 + B1 , B1 ), · · · , (βq + Bq , Bq )
3
(7)
Recalling the contiguous relation (6) for the Fox–Wright function p Ψq (α1 + 1) which is in the right-hand side of the above result, the left-hand side of the above identity can be simplified as: (α1 + 2, A1 ), · · · , (αp , Ap ) (α1 , A1 ), · · · , (αp , Ap ) z = (α1 )2 p Ψq z p Ψq (β1 , B1 ), · · · , (βq , Bq ) (β1 , B1 ), · · · , (βq , Bq ) (α1 + A1 , A1 ), · · · , (αp + Ap , Ap ) +A1 z(α1 + 1)p Ψq z (β1 + B1 , B1 ), · · · , (βq + Bq , Bq ) (α1 + 1 + A1 , A1 ), · · · , (αp + Ap , Ap ) +A1 z p Ψq z , (β1 + B1 , B1 ), · · · , (βq + Bq , Bq ) which is just the contiguous relation (3) with k = 2. Iterating this relation k–times, and after some simplification, we readily obtain (3) in this theorem. Applying the contiguous relation (6) and the inductive method, we can prove the contiguous relation (4). When k = 1, the relation (4) is true obviously. Suppose that (4) is true for k ≤ n, we just need to prove the truth of (4) when k = n + 1. When k = n, the contiguous relation (4) can be expressed as (α1 + n, A1 ), · · · , (αp , Ap ) z p Ψq (β1 , B1 ), · · · , (βq , Bq ) =
n
(A1 z)i
i=0
(α1 + iA1 )ji (α1 + (i − 1)A1 + 1 + ji )ji−1 · · ·
l+j1 +···+ji =n−i;l,j1 ,··· ,ji ≥0
×(α1 + A1 + i − 1 + ji + ji−1 + · · · + j2 )j1 (α1 + i + ji + ji−1 + · · · + j1 )l (α1 + iA1 , A1 ), · · · , (αp + iAp , Ap ) ×p Ψq z . (β1 + iB1 , B1 ), · · · , (βq + iBq , Bq ) Performing the replacement α1 → α1 + 1 in the above result, we can get (α1 + 1 + n, A1 ), · · · , (αp , Ap ) z p Ψq (β1 , B1 ), · · · , (βq , Bq ) =
n i=0
(A1 z)i
(α1 + 1 + iA1 )ji (α1 + (i − 1)A1 + 2 + ji )ji−1 · · ·
l+j1 +···+ji =n−i;l,j1 ,··· ,ji ≥0
×(α1 + A1 + i + ji + ji−1 + · · · + j2 )j1 (α1 + 1 + i + ji + ji−1 + · · · + j1 )l (α1 + 1 + iA1 , A1 ), · · · , (αp + iAp , Ap ) ×p Ψq z . (β1 + iB1 , B1 ), · · · , (βq + iBq , Bq )
(8)
Obviously, the p Ψq –function which is on the right-hand side of the above relation can be evaluated by performing the replacements α1 → α1 + iA1 , · · · , αp → αp + iAp , β1 → β1 + iB1 , · · · , and βq → βq +iBq in the contiguous relation (6). Then the Fox–Wright function p Ψq (α1 +1+n)
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X. Wang
on the left-hand side of the above relation can be evaluated as follows: p Ψq
=
(α1 + 1 + n, A1 ), · · · , (αp , Ap ) z (β1 , B1 ), · · · , (βq , Bq )
n
(A1 z)i
i=0
(α1 + 1 + iA1 )ji (α1 + (i − 1)A1 + 2 + ji )ji−1 · · ·
l+j1 +···+ji =n−i;l,j1 ,··· ,ji ≥0
×(α1 + A1 + i + ji + ji−1 + · · · + j2 )j1 (α1 + 1 + i + ji + ji−1 + · · · + j1 )l
(α1 + iA1 , A1 ), · · · , (αp + iAp , Ap ) × (α1 + iA1 ) p Ψq z (β1 + iB1 , B1 ), · · · , (βq + iBq , Bq ) (α1 + (i + 1)A1 , A1 ), · · · , (αp + (i + 1)Ap , Ap ) +A1 z p Ψq z (β1 + (i + 1)B1 , B1 ), · · · , (βq + (i + 1)Bq , Bq ) =
n+1
(A1 z)i
i=0
(α1 + iA1 )ji +1 (α1 + (i − 1)A1 + 2 + ji )ji−1 · · ·
l+j1 +···+(ji +1)=1+n−i;l,j1 ,··· ,ji ≥0
×(α1 + A1 + i + ji + ji−1 + · · · + j2 )j1 (α1 + 1 + i + ji + ji−1 + · · · + j1 )l + (α1 + (i − 1)A1 + 1)ji−1 · · · l+j1 +···+ji−1 =n−(i−1);l,j1 ,··· ,ji−1 ≥0;ji =0
×(α1 + A1 + (i − 1) + ji−1 + · · · + j2 )j1 (α1 + i + ji−1 + · · · + j1 )l × p Ψq
(α1 + iA1 , A1 ), · · · , (αp + iAp , Ap ) z . (β1 + iB1 , B1 ), · · · , (βq + iBq , Bq )
It is obvious that, on the right-hand side of the above identity, the first summation in the {} means that the factorial (α1 + iA1 ) is always present, and the second summation does not have the factorial (α1 + iA1 ). Then we have the following result: p Ψq
=
(α1 + 1 + n, A1 ), · · · , (αp , Ap ) z (β1 , B1 ), · · · , (βq , Bq )
n+1 i=0
(A1 z)i
(α1 + iA1 )ji (α1 + (i − 1)A1 + 1 + ji )ji−1 · · ·
l+j1 +···+ji =1+n−i;l,j1 ,··· ,ji ≥0
×(α1 + A1 + i − 1 + ji + ji−1 + · · · + j2 )j1 (α1 + i + ji + ji−1 + · · · + j1 )l (α1 + iA1 , A1 ), · · · , (αp + iAp , Ap ) ×p Ψq z , (β1 + iB1 , B1 ), · · · , (βq + iBq , Bq )
which is just the expression of the contiguous relation (4) when k = n + 1. This completes the proof of the contiguous relation (4).
Contiguous relations for the Fox–Wright function
5
Theorem 2. The following contiguous relations of the Fox–Wright function hold true. 1 (α1 − k, A1 ), · · · , (αp , Ap ) (α1 , A1 ), · · · , (αp , Ap ) Ψ Ψ z = z p q p q (β1 , B1 ), · · · , (βq , Bq ) (β1 , B1 ), · · · , (βq , Bq ) (α1 − k)k −A1 z
k i=1
p Ψq
(9)
1 (α1 + A1 − k + i − 1, A1 ), (α2 + A2 , A2 ) · · · , (αp + Ap , Ap ) z ; p Ψq (β1 + B1 , B1 ), · · · , (βq + Bq , Bq ) (α1 − k)i
k (α1 − k, A1 ), · · · , (αp , Ap ) (−A1 z)i z = (β1 , B1 ), · · · , (βq , Bq ) i=0
l+j1 +···+ji =k;l,j1 ,··· ,ji−1 ≥1;ji ≥0
1 (α1 − k)l (α1 + A1 − k − 1 + l)j1 · · · (α1 + iA1 − k − i + l + ji−1 + · · · + j1 )ji (α1 + iA1 − i, A1 ), (α2 + iA2 , A2 ) · · · , (αp + iAp , Ap ) ×p Ψq z . (β1 + iB1 , B1 ), · · · , (βq + iBq , Bq )
(10)
Proof. The proof of this theorem is similar to the proof of Theorem 1. Performing the replacement α1 → α1 − 1 in the contiguous relation (6), we have the new contiguous relation as follows: 1 (α1 − 1, A1 ), · · · , (αp , Ap ) (α1 , A1 ), · · · , (αp , Ap ) z z = p Ψq p Ψq (β1 , B1 ), · · · , (βq , Bq ) (β1 , B1 ), · · · , (βq , Bq ) α1 − 1 A1 z (α1 + A1 − 1, A1 ), · · · , (αp + Ap , Ap ) − (11) z . p Ψq (β1 + B1 , B1 ), · · · , (βq + Bq , Bq ) α1 − 1 Performing the replacement α1 → α1 − 1 in the above contiguous relation, we have 1 (α1 − 2, A1 ), · · · , (αp , Ap ) (α1 − 1, A1 ), · · · , (αp , Ap ) Ψ Ψ z = p q p q (β1 , B1 ), · · · , (βq , Bq ) (β1 , B1 ), · · · , (βq , Bq ) α1 − 2 A1 z (α1 + A1 − 2, A1 ), · · · , (αp + Ap , Ap ) − Ψ z . p q (β1 + B1 , B1 ), · · · , (βq + Bq , Bq ) α1 − 2
z
Applying the contiguous relation (11) to the p Ψq (α1 − 1) of the above identity, the left-hand side of the above relation can be expressed as 1 (α1 − 2, A1 ), · · · , (αp , Ap ) (α1 , A1 ), · · · , (αp , Ap ) z = z p Ψq p Ψq (β1 , B1 ), · · · , (βq , Bq ) (β1 , B1 ), · · · , (βq , Bq ) (α1 − 2)2 A1 z (α1 + A1 − 1, A1 ), · · · , (αp , Ap ) = z p Ψq (β1 + B1 , B1 ), · · · , (βq + Bq , Bq ) (α1 − 2)2 A1 z (α1 + A1 − 2, A1 ), · · · , (αp + Ap , Ap ) − z . p Ψq (β1 + B1 , B1 ), · · · , (βq + Bq , Bq ) α1 − 2 Iterating this computation k–times, we easily get the first contiguous relation (9) in this theorem by relation (11). By the inductive method and the contiguous relation (11), we can get the contiguous relation (10) similarly as in the proof of (4) in Theorem 1. We omit the details which interested readers can easily do by themselves.
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3. Contiguous relations with differing denominator parameters of the Fox–Wright function In this section, we present two theorems providing contiguous relations with differing denominator parameters for the Fox–Wright function. Theorem 3. The following contiguous relations of the Fox–Wright function hold true. (α1 , A1 ), · · · , (αp , Ap ) (α1 , A1 ), · · · , (αp , Ap ) Ψ − k) Ψ z = (β z p q 1 k p q (β1 − k, B1 ), · · · , (βq , Bq ) (β1 , B1 ), · · · , (βq , Bq ) +B1 z
k−1
(β1 − k)k−1−i p Ψq
i=0
p Ψq
(α1 + A1 , A1 ), (α2 + A2 , A2 ) · · · , (αp + Ap , Ap ) z ; (12) (β1 + B1 − i, B1 ), · · · , (βq + Bq , Bq )
k (α1 , A1 ), · · · , (αp , Ap ) (B1 z)i z = (β1 − k, B1 ), · · · , (βq , Bq ) i=0
(β1 − k)l
l+j1 +···+ji =k−i;l,j1 ,··· ,ji ≥0
×(β1 + B1 − k + 1 + l)j1 · · · (β1 + iB1 − k + i + l + j1 + · · · + ji−1 )ji (α1 + iA1 , A1 ), · · · , (αp + iAp , Ap ) ×p Ψq z . (β1 + iB1 , B1 ), · · · , (βq + iBq , Bq )
(13)
Proof. Recalling the relation (1) of the Γ–function, we have 1 β 1 − 1 + B1 k β1 − 1 B1 k = = + . Γ(β1 − 1 + B1 k) Γ(β1 + B1 k) Γ(β1 + B1 k) Γ(β1 + B1 k)
(14)
By the above relation and the definition (2), we can easily get the contiguous relation of the Fox–Wright function p Ψq (β1 − 1) as follows: (α1 , A1 ), · · · , (αp , Ap ) (α1 , A1 ), · · · , (αp , Ap ) z = (β Ψ − 1) Ψ z p q 1 p q (β1 − 1, B1 ), · · · , (βq , Bq ) (β1 , B1 ), · · · , (βq , Bq ) (α1 + A1 , A1 ), · · · , (αp + Ap , Ap ) +B1 z p Ψq (15) z . (β1 + B1 , B1 ), · · · , (βq + Bq , Bq ) Performing the replacement β1 → β1 − 1 in the above contiguous relation, we have (α1 , A1 ), · · · , (αp , Ap ) (α1 , A1 ), · · · , (αp , Ap ) Ψ − 2) Ψ z = (β z p q 1 p q (β1 − 2, B1 ), · · · , (βq , Bq ) (β1 − 1, B1 ), · · · , (βq , Bq ) (α1 + A1 , A1 ), · · · , (αp + Ap , Ap ) +B1 z p Ψq z . (β1 − 1 + B1 , B1 ), · · · , (βq + Bq , Bq ) Applying contiguous relation (15) to the p Ψq (β1 − 1) of the above identity, the left-hand side of the above result can be expressed as (α1 , A1 ), · · · , (αp , Ap ) (α1 , A1 ), · · · , (αp , Ap ) z = (β1 − 2)2 p Ψq z p Ψq (β1 − 2, B1 ), · · · , (βq , Bq ) (β1 − 1, B1 ), · · · , (βq , Bq ) +B1 z(β1 − 2) p Ψq +B1 z p Ψq
(α1 + A1 , A1 ), · · · , (αp + Ap , Ap ) z (β1 + B1 , B1 ), · · · , (βq + Bq , Bq )
(α1 + A1 , A1 ), · · · , (αp + Ap , Ap ) z . (β1 − 1 + B1 , B1 ), · · · , (βq + Bq , Bq )
Contiguous relations for the Fox–Wright function
7
Iterating this evaluation k–times and after some simplification, the contiguous relation of (12) can be found easily. By the contiguous relation (15), we can prove the second contiguous relation (16) in this theorem by the inductive method. Obviously, the contiguous relation (16) is true for k = 0. Suppose the relation (16) is true for k ≤ n, we just need to prove the truth of k = n + 1. When k = n, the contiguous relation (16) can be expressed as: p Ψq
=
(α1 , A1 ), · · · , (αp , Ap ) z (β1 − n, B1 ), · · · , (βq , Bq )
n
(B1 z)i
i=0
(β1 − n)l (β1 + B1 − n + 1 + l)j1
l+j1 +···+ji =n−i;l,j1 ,··· ,ji ≥0
×(β1 + 2B1 − n + 2 + l + j1 )j2 · · · (β1 + iB1 + i − n + l + j1 + · · · + ji−1 )ji ×p Ψq
(α1 + iA1 , A1 ), · · · , (αp + iAp , Ap ) z . (β1 + iB1 , B1 ), · · · , (βq + iBq , Bq )
(16)
Performing the replacement β1 → β1 − n in the contiguous relation (15), we have (α1 , A1 ), · · · , (αp , Ap ) z (β1 − n − 1, B1 ), · · · , (βq , Bq ) (α1 , A1 ), · · · , (αp , Ap ) = (β1 − n − 1) p Ψq z (β1 − n, B1 ), · · · , (βq , Bq )
p Ψq
+B1 z p Ψq
(α1 + A1 , A1 ), · · · , (αp + Ap , Ap ) z . (β1 − n + B1 , B1 ), · · · , (βq + Bq , Bq )
(17)
Perform the replacement β1 → β1 +B1 in (16) directly, the Fox–Wright function in the left-hand side of the above identity can be simplified as follows: p Ψq
=
(α1 , A1 ), · · · , (αp , Ap ) z (β1 − 1 − n, B1 ), · · · , (βq , Bq )
n i=0
(B1 z)i
(β1 − n − 1)l+1 (β1 + B1 − n + 1 + l)j1
l+j1 +···+ji =n−i;l,j1 ,··· ,ji ≥0
×(β1 + 2B1 − n + 2 + l + j1 )j2 · · · (β1 + iB1 + i − n + l + j1 + · · · + ji−1 )ji (α1 + iA1 , A1 ), · · · , (αp + iAp , Ap ) ×p Ψq z (β1 + iB1 , B1 ), · · · , (βq + iBq , Bq ) n + (B1 z)i+1 (β1 + B1 − n)l (β1 + 2B1 − n + 1 + l)j1 i=0
l+j1 +···+ji =n−i;l,j1 ,··· ,ji ≥0
×(β1 + 3B1 − n + 2 + l + j1 )j2 · · · (β1 + (i + 1)B1 + i − n + l + j1 + · · · + ji−1 )ji (α1 + (i + 1)A1 , A1 ), · · · , (αp + (i + 1)Ap , Ap ) ×p Ψq z . (β1 + (i + 1)B1 , B1 ), · · · , (βq + (i + 1)Bq , Bq )
(18)
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X. Wang
Comparing the coefficient of (B1 z)i of the above identity, we have (α1 , A1 ), · · · , (αp , Ap ) Ψ z p q (β1 − 1 − n, B1 ), · · · , (βq , Bq ) =
n
(B1 z)i
i=0
(β1 − n − 1)l+1 (β1 + B1 − n + 1 + l)j1
l+1+j1 +···+ji =1+n−i;l,j1 ,··· ,ji ≥0
×(β1 + 2B1 − n + 2 + l + j1 )j2 · · · (β1 + iB1 + i − n + l + j1 + · · · + ji−1 )ji + (β1 + B1 − n)l (β1 + 2B1 − n + 1 + l)j1 l+j1 +···+ji−1 =n−(i−1);l,j1 ,··· ,ji−1 ≥0
×(β1 + 3B1 − n + 2 + l + j1 )j2 · · · (β1 + iB1 + i − n − 1 + l + j1 + · · · + ji−2 )ji −1 ×p Ψq
(α1 + iA1 , A1 ), · · · , (αp + iAp , Ap ) z . (β1 + iB1 , B1 ), · · · , (βq + iBq , Bq )
(19)
In the above identity, the first summation in the {} means that the factor (β1 − n − 1) is always present, and the second summation in the {} does not have the factor (β1 − n − 1). After some simplification, we have (α1 , A1 ), · · · , (αp , Ap ) z p Ψq (β1 − n − 1, B1 ), · · · , (βq , Bq ) =
n+1
(B1 z)i
i=0
(β1 − n − 1)l (β1 + B1 − n + l)j1
l+j1 +···+ji =1+n−i;l,j1 ,··· ,ji ≥0
×(β1 + 2B1 − n + 1 + l + j1 )j2 · · · (β1 + iB1 + i − n − 1 + l + j1 + · · · + ji−1 )ji (α1 + iA1 , A1 ), · · · , (αp + iAp , Ap ) ×p Ψq z , (β1 + iB1 , B1 ), · · · , (βq + iBq , Bq )
(20)
which is just the contiguous relation (16) with k = n + 1. This proves the truth of the identity (16). Now, the proof of this theorem is completed. Theorem 4. The following contiguous relation of the Fox–Wright function hold true. 1 (α1 , A1 ), · · · , (αp , Ap ) (α1 , A1 ), · · · , (αp , Ap ) z = z p Ψq p Ψq (β1 + k, B1 ), · · · , (βq , Bq ) (β1 , B1 ), · · · , (βq , Bq ) (β1 )k k (β1 )k (α1 + A1 , A1 ), (α2 + A2 , A2 ) · · · , (αp + Ap , Ap ) −B1 z z ; p Ψq (β1 + B1 + i, B1 ), (β2 + B2 , B2 ) · · · , (βq + Bq , Bq ) (β1 )i−1 i=1 (α1 , A1 ), · · · , (αp , Ap ) z p Ψq (β1 + k, B1 ), · · · , (βq , Bq ) =
k i=0
(−B1 z)i
l+j1 +···+ji =k;l,j1 ,··· ,ji−1 ≥1;ji ≥0
(21)
1 (β1 + i(B1 + 1))ji
1 (β1 + (i − 1)(B1 + 1) + ji )ji−1 · · · (β1 + B1 + 1 + ji + · · · + j2 )j1 (β1 + ji + · · · + j1 )l (α1 + iA1 , A1 ), · · · , (αp + iAp , Ap ) ×p Ψq (22) z . (β1 + iB1 + i, B1 ), · · · , (βq + iBq , Bq ) ×
Contiguous relations for the Fox–Wright function
9
Proof. Performing the replacement β1 → β1 + 1 in the contiguous relation (15), we get the new contiguous relation as: 1 (α1 , A1 ), · · · , (αp , Ap ) (α1 , A1 ), · · · , (αp , Ap ) z = z p Ψq (β1 + 1, B1 ), · · · , (βq , Bq ) (β1 , B1 ), · · · , (βq , Bq ) β1 B1 z (α1 + A1 , A1 ), · · · , (αp + Ap , Ap ) − z . p Ψq (β1 + 1 + B1 , B1 ), · · · , (βq + Bq , Bq ) β1
p Ψq
(23)
By the above contiguous relation, the two contiguous relations of this theorem can be proved by the similarly method as the proof of Theorem 3. Here, we will present with no details.
4. The proof of the Hagen–Rothe convolution In this section, we will prove the famous Hagen–Rothe convolution [10] by the formulas we have obtained in our paper. The famous Hagen–Rothe convolution can be expressed as the following three identities: n k=0
a+c c + bk a − bk a ; = n n−k k a − bk
(24)
a+c ; n
(25)
n a − bk c + bn c + bk k=0 n k=0
k
c + bk
n−k
=
a+c a − bk c + bn c + bk a . = n k a − bk c + bk n − k
(26)
Proof. The Hagen–Rothe convolution (25) can be obtained by reversing the summation order of the identity (24). The identity (26) can be proved by the linear combination of the identities (24) and (25). So we just need to prove the correctness of the convolution (24). Obviously, the identity (24) can be rewritten as 2 Ψ3
1 a + c (a, −b), (1 + c, b) . 1 = (n + 1, −1), (1 + a, −b − 1), (1 + c − n, 1 + b) n a
(27)
Here, we will prove the identity (27) by the inductive method. When n = 0, The result (27) is correct obviously. Suppose that the result (27) is true when n = m. Here, we just need to prove the truth of (27) with n = m + 1. By the contiguous relation (15), we have
(a, −b), (1 + c, b) 1 (m + 2, −1), (1 + a, −b − 1), (c − m, 1 + b) (a, −b), (1 + c, b) = (c − m) 2 Ψ3 1 (m + 2, −1), (1 + a, −b − 1), (1 + c − m, 1 + b) (a − b, −b), (1 + b + c, b) +(1 + b) 2 Ψ3 1 . (m + 1, −1), (a − b, −b − 1), (2 + b + c − m, 1 + b) 2 Ψ3
10
X. Wang
Applying the contiguous relation (23) on the first Fox–Wright function in the right-hand side of the above identity, we just arrive at
(a, −b), (1 + c, b) 1 2 Ψ3 (m + 2, −1), (1 + a, −b − 1), (c − m, 1 + b)
1 (a, −b), (1 + c, b) = (c − m) Ψ 1 2 3 (m + 1, −1), (1 + a, −b − 1), (1 + c − m, 1 + b) m+1 1 (a − b, −b), (1 + b + c, b) + Ψ 1 2 3 (m + 1, −1), (a − b, −b − 1), (2 + b + c − m, 1 + b) m+1 (a − b, −b), (1 + b + c, b) +(1 + b)2 Ψ3 1 (m + 1, −1), (a − b, −b − 1), (2 + b + c − m, 1 + b) c−m (a, −b), (1 + c, b) = Ψ 1 2 3 (m + 1, −1), (1 + a, −b − 1), (1 + c − m, 1 + b) m+1 1 + b + c + bm (a − b, −b), (1 + b + c, b) + 1 . 2 Ψ3 (m + 1, −1), (a − b, −b − 1), (2 + b + c − m, 1 + b) m+1 It is obvious that the first 2 Ψ3 -series in the right-hand side of the above identity is just the identity (27) with n = m. Reversing the summation order of the second 2 Ψ3 -series in the right-hand side of the above identity, we have 2 Ψ3
=
(a − b, −b), (1 + b + c, b) 1 (m + 1, −1), (a − b, −b − 1), (2 + b + c − m, 1 + b)
m k=0
=
m k=0
= 2 Ψ3
Γ(a − b − bk)Γ(1 + b + c + bk) 1 Γ(m + 1 − k)Γ(a − b − bk − k)Γ(2 + b + c − m + bk + k) k! Γ(a − b − bm + bk)Γ(1 + b + c + bm − bk) 1 Γ(m + 1 − k)Γ(a − b − bm − m + bk + k)Γ(2 + b + c + bm − bk − k) k!
(a − b − bm, b), (1 + b + c + bm, −b) 1 , (m + 1, −1), (a − b − bm − m, b + 1), (2 + b + c + bm, −1 − b)
where we have performed the replacement k → m − k in the second equality. It is obvious that the 2 Ψ3 -series in the right-hand side of the above identity is just the Hagen-Rothe formula (27) with the replacement a → 1 + b + c + bm and c → a − b − bm − 1. Then we can calculate that
(a, −b), (1 + c, b) 1 (m + 2, −1), (1 + a, −b − 1), (c − m, 1 + b)
a+c 1 + b + c + bm a+c c−m 1 + = m m a(1 + m) 1+m 1 + b + c + bm
1 a+c = , a m+1 2 Ψ3
which is just the identity (27) with n = m + 1. This completes the proof of the Hagen–Rothe convolutions.
Contiguous relations for the Fox–Wright function
11
Acknowledgement This work is supported by National Natural Science Foundations of China (11661032). References [1] Andrews GE, Askey R, Roy R. Special Functions. Cambridge Univ. Press, Cambridge, 2000. [2] Brychkov YuA, Saad N. Some formulas for the Appell function F1 (a, b, b ; c; w, z). Integral Transforms and Special Functions. 2012; 23(11): 793–802. [3] Brychkov YuA, Saad N. On some formulas for the Appell function F2 (a, b, b ; c, c ; w, z). Integral Transforms and Special Functions. 2014; 25(2): 111–123. [4] Brychkov YuA, Saad N. On some formulas for the Appell function F3 (a, a , b, b ; c; w, z). Integral Transforms and Special Functions. 2015; 26(11): 910–923. [5] Brychkov YuA, Saad N. On some formulas for the Appell function F4 (a, b; c, c ; w, z). Integral Transforms and Special Functions. 2017; 28(9): 629–644. [6] Chu W, Wang X. Abel’s method on summation by parts and hypergeometric contiguous relations. Integral Transforms and Special Functions. 2007; 28(11): 771–807. [7] Chu W, Wang X. Summation formulae on Fox–Wright Ψ-functions. Integral Transforms and Special Functions. 2008; 19(8): 545–561. [8] Craven T, Csorads G. The Fox–Wright functions and Laguerre multiplier sequences. J. Math. Anal. Appl. 2006; 314: 109–125. [9] Darus M, Ibrahim RW. On Ces´ aro means for Fox–Wright functions. J. Math. Stat. 2008; 4: 156–160. [10] Gould HW. Some generalizations of Vandermonde’s convolution. Amer. Math. Month., 1956; 63: 84–91. [11] Fox C. The asymptotic expansion of generalized hypergeometric functions. Proc. Lond. Math. Soc. 1928; 27 (3) (Ser. 2): 389–400. [12] Krattenthaler C, Rivoal T. How can we escape Thomae’s relations? J. Math. Soc. Japan 2006; 58: 183–210. [13] Mainardi F, Pagnini G. The role of the Fox–Wright functions in fractional sub–diffusion of distributed order. J. Comput. Appl. Math. 2007; 207: 245–257. [14] Miller AR, Moskowitz IS. Reduction of a class of Fox–Wright Psi functions for certain rational parameters. Comput. Math. Appl. 1995; 30: 73–82. [15] Miller AR. On the Mellin transform of a product of two Fox–Wright psi functions. J. Phys. A: Math. Gen. 2002; 35: 2275–2281. [16] Opps SB, Saad N, Srivastava HM. Recursion formulas for Appell’s hypergeometric function F2 with some applications to radiation field problem. Applied Mathematics and Computation. 2009; 207: 545–558. [17] Srivastava HM, Karlsson PW. Multiple Gaussian Hypergeometric Series. Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York; 1985. [18] Wang X. Recursion formulas for Appell functions. Integral Transformations and Special Functions. 2012; 23(6): 421–433. [19] Wei C, Gong D, Hao H. A family of summation formulas on the Fox–Wright function. J. Math. Anal. Appl. 2012; 393: 316–327. [20] Whipple JF. A group of generalized hypergeometric series: relations between 120 allied series of type F[a, b, c; d, e]. Proceedings of the London Mathematical Society, 1925; 23(2): 104–C114.. [21] Wright EM. The asymptotic expansion of generalized hypergeometric functions. J. Lond. Math. Soc. 1935; 10: 286–293. [22] Wright EM. The asymptotic expansion of generalized hypergeometric functions. Proc. Lond. Math. Soc. 1940; 46 (3)(Ser. 2): 389–408.
Xiaoxia Wang
PII: DOI: Reference:
S0022-247X(19)30159-3 https://doi.org/10.1016/j.jmaa.2019.02.035 YJMAA 22968
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
6 April 2018
Please cite this article in press as: X. Wang, Contiguous relations for the Fox–Wright function, J. Math. Anal. Appl. (2019), https://doi.org/10.1016/j.jmaa.2019.02.035
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CONTIGUOUS RELATIONS FOR THE FOX–WRIGHT FUNCTION XIAOXIA WANG† Department of Mathematics, Shanghai University, Shanghai 200444, P. R. China
Contiguous relations are a powerful tool for dealing with hypergeometric functions. In this paper, four different types of contiguous relations for the Fox–Wright function are given. We demonstrate the usefulness of our formulas by showing how they can be used to recover the famous Hagen–Rothe convolution. Keywords: Fox–Wright function; Gamma function; contiguous relation; Hagen–Rothe convolution. 2010 Mathematics Subject Classification: Primary 33C20; Secondary 33C99.
1. Introduction The Pochhammer symbol (or the rising shifted factorial, since (1)n = n!) defined, for any complex number α, by ⎧ Γ(α + n) ⎨α(α + 1) · · · (α + n − 1), n ∈ N = 1, 2, 3, · · ·, = (α)n := ⎩1, Γ(α) n = 0, where the Γ–function [1], with the complex numbers x = 0, −1, −2, · · · , being defined as ∞ Γ(x) = ux−1 e−u du, R(x) > 0. 0
The following relation about the Γ–function Γ(x + 1) = xΓ(x),
(1)
will be used frequently in this paper. The Fox–Wright function [11, 17, 21, 22], which is regarded as a generalization of hypergeometric function, is defined as follows: ∞ Πpi=1 Γ(αi + Ai k) z k (α1 , A1 ), · · · , (αp , Ap ) , (2) Ψ z := p q (β1 , B1 ), · · · , (βq , Bq ) Πqj=1 Γ(βj + Bj k) k! k=0
where the coefficients {Ai }i≥1 and {Bj }j≥1 are positive real numbers with 1+
q j=1
Bj −
p i=1
E-mail addresses: [email protected] (X. Wang).
Ai ≥ 0.
2
X. Wang
Recently, many summations for the Fox–Wright function are established in [7, 19]. The importance of the Fox–Wright function can be found in [8, 9, 13, 14, 15]. Recently, Opps et al. [16] established the recursion formulas of Appell’s function F2 . Brychkov and Saad [2, 3, 4, 5] researched Appell’s four functions F1 , F2 , F3 and F4 including the recursion formulas. Many contiguous relations of 3 F2 -series are given by Whipple [20] and Krattenthaler and Rivoal [12]. The author with the partner established many contiguous relations of 3 F2 series by Abel’s method [6] and recursion formulas of Appell’s functions [18]. In this paper, we present four different types of contiguous relations for the Fox–Wright function.
2. Contiguous relations with differing numerator parameters of the Fox–Wright function 2.1. In this section, we first establish the contiguous relation about the numerator α1 , then we get the contiguous formulas about the numerator parameter α1 with two expressions. The contiguous relations about the other numerator parameters can be established by the similarly method. Theorem 1. The following contiguous relations of the Fox–Wright function hold true. (α1 + k, A1 ), · · · , (αp , Ap ) (α1 , A1 ), · · · , (αp , Ap ) z = (α1 )k p Ψq z p Ψq (β1 , B1 ), · · · , (βq , Bq ) (β1 , B1 ), · · · , (βq , Bq ) +A1 z p Ψq
=
i=1
(α1 )i
p Ψq
(α1 + i − 1 + A1 , A1 ), (α2 + A2 , A2 ) · · · , (αp + Ap , Ap ) z ; (3) (β1 + B1 , B1 ), · · · , (βq + Bq , Bq )
(α1 + k, A1 ), · · · , (αp , Ap ) z (β1 , B1 ), · · · , (βq , Bq )
k i=0
k (α1 )k
(A1 z)i
(α1 + iA1 )ji (α1 + (i − 1)A1 + 1 + ji )ji−1 · · ·
l+j1 +···+ji =k−i;l,j1 ,··· ,ji ≥0
×(α1 + A1 + i − 1 + ji + ji−1 + · · · + j2 )j1 (α1 + i + ji + ji−1 + · · · + j1 )l (α1 + iA1 , A1 ), · · · , (αp + iAp , Ap ) ×p Ψq z . (β1 + iB1 , B1 ), · · · , (βq + iBq , Bq ) Proof. From the definition of the Fox–Wright function (2), we have ∞ Γ(α1 + 1 + A1 k)Πpi=2 Γ(αi + Ai k) z k (α1 + 1, A1 ), · · · , (αp , Ap ) . Ψ z = p q (β1 , B1 ), · · · , (βq , Bq ) Πqj=1 Γ(βj + Bj k) k!
(4)
(5)
k=0
Recalling the relation of the Γ–function (1), we get Γ(α1 + 1 + A1 k) = Γ(α1 + A1 k)(α1 + A1 k) = α1 Γ(α1 + A1 k) + A1 kΓ(α1 + A1 k). Then the left-hand side of (5) can be expressed as the following contiguous relation (α1 + 1, A1 ), · · · , (αp , Ap ) (α1 , A1 ), · · · , (αp , Ap ) z = α1 p Ψq z p Ψq (β1 , B1 ), · · · , (βq , Bq ) (β1 , B1 ), · · · , (βq , Bq ) (α1 + A1 , A1 ), · · · , (αp + Ap , Ap ) +A1 z p Ψq z . (β1 + B1 , B1 ), · · · , (βq + Bq , Bq )
(6)
Contiguous relations for the Fox–Wright function
Performing the replacement α1 → α1 + 1 in the above contiguous relation, we have (α1 + 2, A1 ), · · · , (αp , Ap ) (α1 + 1, A1 ), · · · , (αp , Ap ) Ψ + 1) Ψ z = (α z p q 1 p q (β1 , B1 ), · · · , (βq , Bq ) (β1 , B1 ), · · · , (βq , Bq ) (α1 + 1 + A1 , A1 ), · · · , (αp + Ap , Ap ) +A1 z p Ψq z . (β1 + B1 , B1 ), · · · , (βq + Bq , Bq )
3
(7)
Recalling the contiguous relation (6) for the Fox–Wright function p Ψq (α1 + 1) which is in the right-hand side of the above result, the left-hand side of the above identity can be simplified as: (α1 + 2, A1 ), · · · , (αp , Ap ) (α1 , A1 ), · · · , (αp , Ap ) z = (α1 )2 p Ψq z p Ψq (β1 , B1 ), · · · , (βq , Bq ) (β1 , B1 ), · · · , (βq , Bq ) (α1 + A1 , A1 ), · · · , (αp + Ap , Ap ) +A1 z(α1 + 1)p Ψq z (β1 + B1 , B1 ), · · · , (βq + Bq , Bq ) (α1 + 1 + A1 , A1 ), · · · , (αp + Ap , Ap ) +A1 z p Ψq z , (β1 + B1 , B1 ), · · · , (βq + Bq , Bq ) which is just the contiguous relation (3) with k = 2. Iterating this relation k–times, and after some simplification, we readily obtain (3) in this theorem. Applying the contiguous relation (6) and the inductive method, we can prove the contiguous relation (4). When k = 1, the relation (4) is true obviously. Suppose that (4) is true for k ≤ n, we just need to prove the truth of (4) when k = n + 1. When k = n, the contiguous relation (4) can be expressed as (α1 + n, A1 ), · · · , (αp , Ap ) z p Ψq (β1 , B1 ), · · · , (βq , Bq ) =
n
(A1 z)i
i=0
(α1 + iA1 )ji (α1 + (i − 1)A1 + 1 + ji )ji−1 · · ·
l+j1 +···+ji =n−i;l,j1 ,··· ,ji ≥0
×(α1 + A1 + i − 1 + ji + ji−1 + · · · + j2 )j1 (α1 + i + ji + ji−1 + · · · + j1 )l (α1 + iA1 , A1 ), · · · , (αp + iAp , Ap ) ×p Ψq z . (β1 + iB1 , B1 ), · · · , (βq + iBq , Bq ) Performing the replacement α1 → α1 + 1 in the above result, we can get (α1 + 1 + n, A1 ), · · · , (αp , Ap ) z p Ψq (β1 , B1 ), · · · , (βq , Bq ) =
n i=0
(A1 z)i
(α1 + 1 + iA1 )ji (α1 + (i − 1)A1 + 2 + ji )ji−1 · · ·
l+j1 +···+ji =n−i;l,j1 ,··· ,ji ≥0
×(α1 + A1 + i + ji + ji−1 + · · · + j2 )j1 (α1 + 1 + i + ji + ji−1 + · · · + j1 )l (α1 + 1 + iA1 , A1 ), · · · , (αp + iAp , Ap ) ×p Ψq z . (β1 + iB1 , B1 ), · · · , (βq + iBq , Bq )
(8)
Obviously, the p Ψq –function which is on the right-hand side of the above relation can be evaluated by performing the replacements α1 → α1 + iA1 , · · · , αp → αp + iAp , β1 → β1 + iB1 , · · · , and βq → βq +iBq in the contiguous relation (6). Then the Fox–Wright function p Ψq (α1 +1+n)
4
X. Wang
on the left-hand side of the above relation can be evaluated as follows: p Ψq
=
(α1 + 1 + n, A1 ), · · · , (αp , Ap ) z (β1 , B1 ), · · · , (βq , Bq )
n
(A1 z)i
i=0
(α1 + 1 + iA1 )ji (α1 + (i − 1)A1 + 2 + ji )ji−1 · · ·
l+j1 +···+ji =n−i;l,j1 ,··· ,ji ≥0
×(α1 + A1 + i + ji + ji−1 + · · · + j2 )j1 (α1 + 1 + i + ji + ji−1 + · · · + j1 )l
(α1 + iA1 , A1 ), · · · , (αp + iAp , Ap ) × (α1 + iA1 ) p Ψq z (β1 + iB1 , B1 ), · · · , (βq + iBq , Bq ) (α1 + (i + 1)A1 , A1 ), · · · , (αp + (i + 1)Ap , Ap ) +A1 z p Ψq z (β1 + (i + 1)B1 , B1 ), · · · , (βq + (i + 1)Bq , Bq ) =
n+1
(A1 z)i
i=0
(α1 + iA1 )ji +1 (α1 + (i − 1)A1 + 2 + ji )ji−1 · · ·
l+j1 +···+(ji +1)=1+n−i;l,j1 ,··· ,ji ≥0
×(α1 + A1 + i + ji + ji−1 + · · · + j2 )j1 (α1 + 1 + i + ji + ji−1 + · · · + j1 )l + (α1 + (i − 1)A1 + 1)ji−1 · · · l+j1 +···+ji−1 =n−(i−1);l,j1 ,··· ,ji−1 ≥0;ji =0
×(α1 + A1 + (i − 1) + ji−1 + · · · + j2 )j1 (α1 + i + ji−1 + · · · + j1 )l × p Ψq
(α1 + iA1 , A1 ), · · · , (αp + iAp , Ap ) z . (β1 + iB1 , B1 ), · · · , (βq + iBq , Bq )
It is obvious that, on the right-hand side of the above identity, the first summation in the {} means that the factorial (α1 + iA1 ) is always present, and the second summation does not have the factorial (α1 + iA1 ). Then we have the following result: p Ψq
=
(α1 + 1 + n, A1 ), · · · , (αp , Ap ) z (β1 , B1 ), · · · , (βq , Bq )
n+1 i=0
(A1 z)i
(α1 + iA1 )ji (α1 + (i − 1)A1 + 1 + ji )ji−1 · · ·
l+j1 +···+ji =1+n−i;l,j1 ,··· ,ji ≥0
×(α1 + A1 + i − 1 + ji + ji−1 + · · · + j2 )j1 (α1 + i + ji + ji−1 + · · · + j1 )l (α1 + iA1 , A1 ), · · · , (αp + iAp , Ap ) ×p Ψq z , (β1 + iB1 , B1 ), · · · , (βq + iBq , Bq )
which is just the expression of the contiguous relation (4) when k = n + 1. This completes the proof of the contiguous relation (4).
Contiguous relations for the Fox–Wright function
5
Theorem 2. The following contiguous relations of the Fox–Wright function hold true. 1 (α1 − k, A1 ), · · · , (αp , Ap ) (α1 , A1 ), · · · , (αp , Ap ) Ψ Ψ z = z p q p q (β1 , B1 ), · · · , (βq , Bq ) (β1 , B1 ), · · · , (βq , Bq ) (α1 − k)k −A1 z
k i=1
p Ψq
(9)
1 (α1 + A1 − k + i − 1, A1 ), (α2 + A2 , A2 ) · · · , (αp + Ap , Ap ) z ; p Ψq (β1 + B1 , B1 ), · · · , (βq + Bq , Bq ) (α1 − k)i
k (α1 − k, A1 ), · · · , (αp , Ap ) (−A1 z)i z = (β1 , B1 ), · · · , (βq , Bq ) i=0
l+j1 +···+ji =k;l,j1 ,··· ,ji−1 ≥1;ji ≥0
1 (α1 − k)l (α1 + A1 − k − 1 + l)j1 · · · (α1 + iA1 − k − i + l + ji−1 + · · · + j1 )ji (α1 + iA1 − i, A1 ), (α2 + iA2 , A2 ) · · · , (αp + iAp , Ap ) ×p Ψq z . (β1 + iB1 , B1 ), · · · , (βq + iBq , Bq )
(10)
Proof. The proof of this theorem is similar to the proof of Theorem 1. Performing the replacement α1 → α1 − 1 in the contiguous relation (6), we have the new contiguous relation as follows: 1 (α1 − 1, A1 ), · · · , (αp , Ap ) (α1 , A1 ), · · · , (αp , Ap ) z z = p Ψq p Ψq (β1 , B1 ), · · · , (βq , Bq ) (β1 , B1 ), · · · , (βq , Bq ) α1 − 1 A1 z (α1 + A1 − 1, A1 ), · · · , (αp + Ap , Ap ) − (11) z . p Ψq (β1 + B1 , B1 ), · · · , (βq + Bq , Bq ) α1 − 1 Performing the replacement α1 → α1 − 1 in the above contiguous relation, we have 1 (α1 − 2, A1 ), · · · , (αp , Ap ) (α1 − 1, A1 ), · · · , (αp , Ap ) Ψ Ψ z = p q p q (β1 , B1 ), · · · , (βq , Bq ) (β1 , B1 ), · · · , (βq , Bq ) α1 − 2 A1 z (α1 + A1 − 2, A1 ), · · · , (αp + Ap , Ap ) − Ψ z . p q (β1 + B1 , B1 ), · · · , (βq + Bq , Bq ) α1 − 2
z
Applying the contiguous relation (11) to the p Ψq (α1 − 1) of the above identity, the left-hand side of the above relation can be expressed as 1 (α1 − 2, A1 ), · · · , (αp , Ap ) (α1 , A1 ), · · · , (αp , Ap ) z = z p Ψq p Ψq (β1 , B1 ), · · · , (βq , Bq ) (β1 , B1 ), · · · , (βq , Bq ) (α1 − 2)2 A1 z (α1 + A1 − 1, A1 ), · · · , (αp , Ap ) = z p Ψq (β1 + B1 , B1 ), · · · , (βq + Bq , Bq ) (α1 − 2)2 A1 z (α1 + A1 − 2, A1 ), · · · , (αp + Ap , Ap ) − z . p Ψq (β1 + B1 , B1 ), · · · , (βq + Bq , Bq ) α1 − 2 Iterating this computation k–times, we easily get the first contiguous relation (9) in this theorem by relation (11). By the inductive method and the contiguous relation (11), we can get the contiguous relation (10) similarly as in the proof of (4) in Theorem 1. We omit the details which interested readers can easily do by themselves.
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X. Wang
3. Contiguous relations with differing denominator parameters of the Fox–Wright function In this section, we present two theorems providing contiguous relations with differing denominator parameters for the Fox–Wright function. Theorem 3. The following contiguous relations of the Fox–Wright function hold true. (α1 , A1 ), · · · , (αp , Ap ) (α1 , A1 ), · · · , (αp , Ap ) Ψ − k) Ψ z = (β z p q 1 k p q (β1 − k, B1 ), · · · , (βq , Bq ) (β1 , B1 ), · · · , (βq , Bq ) +B1 z
k−1
(β1 − k)k−1−i p Ψq
i=0
p Ψq
(α1 + A1 , A1 ), (α2 + A2 , A2 ) · · · , (αp + Ap , Ap ) z ; (12) (β1 + B1 − i, B1 ), · · · , (βq + Bq , Bq )
k (α1 , A1 ), · · · , (αp , Ap ) (B1 z)i z = (β1 − k, B1 ), · · · , (βq , Bq ) i=0
(β1 − k)l
l+j1 +···+ji =k−i;l,j1 ,··· ,ji ≥0
×(β1 + B1 − k + 1 + l)j1 · · · (β1 + iB1 − k + i + l + j1 + · · · + ji−1 )ji (α1 + iA1 , A1 ), · · · , (αp + iAp , Ap ) ×p Ψq z . (β1 + iB1 , B1 ), · · · , (βq + iBq , Bq )
(13)
Proof. Recalling the relation (1) of the Γ–function, we have 1 β 1 − 1 + B1 k β1 − 1 B1 k = = + . Γ(β1 − 1 + B1 k) Γ(β1 + B1 k) Γ(β1 + B1 k) Γ(β1 + B1 k)
(14)
By the above relation and the definition (2), we can easily get the contiguous relation of the Fox–Wright function p Ψq (β1 − 1) as follows: (α1 , A1 ), · · · , (αp , Ap ) (α1 , A1 ), · · · , (αp , Ap ) z = (β Ψ − 1) Ψ z p q 1 p q (β1 − 1, B1 ), · · · , (βq , Bq ) (β1 , B1 ), · · · , (βq , Bq ) (α1 + A1 , A1 ), · · · , (αp + Ap , Ap ) +B1 z p Ψq (15) z . (β1 + B1 , B1 ), · · · , (βq + Bq , Bq ) Performing the replacement β1 → β1 − 1 in the above contiguous relation, we have (α1 , A1 ), · · · , (αp , Ap ) (α1 , A1 ), · · · , (αp , Ap ) Ψ − 2) Ψ z = (β z p q 1 p q (β1 − 2, B1 ), · · · , (βq , Bq ) (β1 − 1, B1 ), · · · , (βq , Bq ) (α1 + A1 , A1 ), · · · , (αp + Ap , Ap ) +B1 z p Ψq z . (β1 − 1 + B1 , B1 ), · · · , (βq + Bq , Bq ) Applying contiguous relation (15) to the p Ψq (β1 − 1) of the above identity, the left-hand side of the above result can be expressed as (α1 , A1 ), · · · , (αp , Ap ) (α1 , A1 ), · · · , (αp , Ap ) z = (β1 − 2)2 p Ψq z p Ψq (β1 − 2, B1 ), · · · , (βq , Bq ) (β1 − 1, B1 ), · · · , (βq , Bq ) +B1 z(β1 − 2) p Ψq +B1 z p Ψq
(α1 + A1 , A1 ), · · · , (αp + Ap , Ap ) z (β1 + B1 , B1 ), · · · , (βq + Bq , Bq )
(α1 + A1 , A1 ), · · · , (αp + Ap , Ap ) z . (β1 − 1 + B1 , B1 ), · · · , (βq + Bq , Bq )
Contiguous relations for the Fox–Wright function
7
Iterating this evaluation k–times and after some simplification, the contiguous relation of (12) can be found easily. By the contiguous relation (15), we can prove the second contiguous relation (16) in this theorem by the inductive method. Obviously, the contiguous relation (16) is true for k = 0. Suppose the relation (16) is true for k ≤ n, we just need to prove the truth of k = n + 1. When k = n, the contiguous relation (16) can be expressed as: p Ψq
=
(α1 , A1 ), · · · , (αp , Ap ) z (β1 − n, B1 ), · · · , (βq , Bq )
n
(B1 z)i
i=0
(β1 − n)l (β1 + B1 − n + 1 + l)j1
l+j1 +···+ji =n−i;l,j1 ,··· ,ji ≥0
×(β1 + 2B1 − n + 2 + l + j1 )j2 · · · (β1 + iB1 + i − n + l + j1 + · · · + ji−1 )ji ×p Ψq
(α1 + iA1 , A1 ), · · · , (αp + iAp , Ap ) z . (β1 + iB1 , B1 ), · · · , (βq + iBq , Bq )
(16)
Performing the replacement β1 → β1 − n in the contiguous relation (15), we have (α1 , A1 ), · · · , (αp , Ap ) z (β1 − n − 1, B1 ), · · · , (βq , Bq ) (α1 , A1 ), · · · , (αp , Ap ) = (β1 − n − 1) p Ψq z (β1 − n, B1 ), · · · , (βq , Bq )
p Ψq
+B1 z p Ψq
(α1 + A1 , A1 ), · · · , (αp + Ap , Ap ) z . (β1 − n + B1 , B1 ), · · · , (βq + Bq , Bq )
(17)
Perform the replacement β1 → β1 +B1 in (16) directly, the Fox–Wright function in the left-hand side of the above identity can be simplified as follows: p Ψq
=
(α1 , A1 ), · · · , (αp , Ap ) z (β1 − 1 − n, B1 ), · · · , (βq , Bq )
n i=0
(B1 z)i
(β1 − n − 1)l+1 (β1 + B1 − n + 1 + l)j1
l+j1 +···+ji =n−i;l,j1 ,··· ,ji ≥0
×(β1 + 2B1 − n + 2 + l + j1 )j2 · · · (β1 + iB1 + i − n + l + j1 + · · · + ji−1 )ji (α1 + iA1 , A1 ), · · · , (αp + iAp , Ap ) ×p Ψq z (β1 + iB1 , B1 ), · · · , (βq + iBq , Bq ) n + (B1 z)i+1 (β1 + B1 − n)l (β1 + 2B1 − n + 1 + l)j1 i=0
l+j1 +···+ji =n−i;l,j1 ,··· ,ji ≥0
×(β1 + 3B1 − n + 2 + l + j1 )j2 · · · (β1 + (i + 1)B1 + i − n + l + j1 + · · · + ji−1 )ji (α1 + (i + 1)A1 , A1 ), · · · , (αp + (i + 1)Ap , Ap ) ×p Ψq z . (β1 + (i + 1)B1 , B1 ), · · · , (βq + (i + 1)Bq , Bq )
(18)
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X. Wang
Comparing the coefficient of (B1 z)i of the above identity, we have (α1 , A1 ), · · · , (αp , Ap ) Ψ z p q (β1 − 1 − n, B1 ), · · · , (βq , Bq ) =
n
(B1 z)i
i=0
(β1 − n − 1)l+1 (β1 + B1 − n + 1 + l)j1
l+1+j1 +···+ji =1+n−i;l,j1 ,··· ,ji ≥0
×(β1 + 2B1 − n + 2 + l + j1 )j2 · · · (β1 + iB1 + i − n + l + j1 + · · · + ji−1 )ji + (β1 + B1 − n)l (β1 + 2B1 − n + 1 + l)j1 l+j1 +···+ji−1 =n−(i−1);l,j1 ,··· ,ji−1 ≥0
×(β1 + 3B1 − n + 2 + l + j1 )j2 · · · (β1 + iB1 + i − n − 1 + l + j1 + · · · + ji−2 )ji −1 ×p Ψq
(α1 + iA1 , A1 ), · · · , (αp + iAp , Ap ) z . (β1 + iB1 , B1 ), · · · , (βq + iBq , Bq )
(19)
In the above identity, the first summation in the {} means that the factor (β1 − n − 1) is always present, and the second summation in the {} does not have the factor (β1 − n − 1). After some simplification, we have (α1 , A1 ), · · · , (αp , Ap ) z p Ψq (β1 − n − 1, B1 ), · · · , (βq , Bq ) =
n+1
(B1 z)i
i=0
(β1 − n − 1)l (β1 + B1 − n + l)j1
l+j1 +···+ji =1+n−i;l,j1 ,··· ,ji ≥0
×(β1 + 2B1 − n + 1 + l + j1 )j2 · · · (β1 + iB1 + i − n − 1 + l + j1 + · · · + ji−1 )ji (α1 + iA1 , A1 ), · · · , (αp + iAp , Ap ) ×p Ψq z , (β1 + iB1 , B1 ), · · · , (βq + iBq , Bq )
(20)
which is just the contiguous relation (16) with k = n + 1. This proves the truth of the identity (16). Now, the proof of this theorem is completed. Theorem 4. The following contiguous relation of the Fox–Wright function hold true. 1 (α1 , A1 ), · · · , (αp , Ap ) (α1 , A1 ), · · · , (αp , Ap ) z = z p Ψq p Ψq (β1 + k, B1 ), · · · , (βq , Bq ) (β1 , B1 ), · · · , (βq , Bq ) (β1 )k k (β1 )k (α1 + A1 , A1 ), (α2 + A2 , A2 ) · · · , (αp + Ap , Ap ) −B1 z z ; p Ψq (β1 + B1 + i, B1 ), (β2 + B2 , B2 ) · · · , (βq + Bq , Bq ) (β1 )i−1 i=1 (α1 , A1 ), · · · , (αp , Ap ) z p Ψq (β1 + k, B1 ), · · · , (βq , Bq ) =
k i=0
(−B1 z)i
l+j1 +···+ji =k;l,j1 ,··· ,ji−1 ≥1;ji ≥0
(21)
1 (β1 + i(B1 + 1))ji
1 (β1 + (i − 1)(B1 + 1) + ji )ji−1 · · · (β1 + B1 + 1 + ji + · · · + j2 )j1 (β1 + ji + · · · + j1 )l (α1 + iA1 , A1 ), · · · , (αp + iAp , Ap ) ×p Ψq (22) z . (β1 + iB1 + i, B1 ), · · · , (βq + iBq , Bq ) ×
Contiguous relations for the Fox–Wright function
9
Proof. Performing the replacement β1 → β1 + 1 in the contiguous relation (15), we get the new contiguous relation as: 1 (α1 , A1 ), · · · , (αp , Ap ) (α1 , A1 ), · · · , (αp , Ap ) z = z p Ψq (β1 + 1, B1 ), · · · , (βq , Bq ) (β1 , B1 ), · · · , (βq , Bq ) β1 B1 z (α1 + A1 , A1 ), · · · , (αp + Ap , Ap ) − z . p Ψq (β1 + 1 + B1 , B1 ), · · · , (βq + Bq , Bq ) β1
p Ψq
(23)
By the above contiguous relation, the two contiguous relations of this theorem can be proved by the similarly method as the proof of Theorem 3. Here, we will present with no details.
4. The proof of the Hagen–Rothe convolution In this section, we will prove the famous Hagen–Rothe convolution [10] by the formulas we have obtained in our paper. The famous Hagen–Rothe convolution can be expressed as the following three identities: n k=0
a+c c + bk a − bk a ; = n n−k k a − bk
(24)
a+c ; n
(25)
n a − bk c + bn c + bk k=0 n k=0
k
c + bk
n−k
=
a+c a − bk c + bn c + bk a . = n k a − bk c + bk n − k
(26)
Proof. The Hagen–Rothe convolution (25) can be obtained by reversing the summation order of the identity (24). The identity (26) can be proved by the linear combination of the identities (24) and (25). So we just need to prove the correctness of the convolution (24). Obviously, the identity (24) can be rewritten as 2 Ψ3
1 a + c (a, −b), (1 + c, b) . 1 = (n + 1, −1), (1 + a, −b − 1), (1 + c − n, 1 + b) n a
(27)
Here, we will prove the identity (27) by the inductive method. When n = 0, The result (27) is correct obviously. Suppose that the result (27) is true when n = m. Here, we just need to prove the truth of (27) with n = m + 1. By the contiguous relation (15), we have
(a, −b), (1 + c, b) 1 (m + 2, −1), (1 + a, −b − 1), (c − m, 1 + b) (a, −b), (1 + c, b) = (c − m) 2 Ψ3 1 (m + 2, −1), (1 + a, −b − 1), (1 + c − m, 1 + b) (a − b, −b), (1 + b + c, b) +(1 + b) 2 Ψ3 1 . (m + 1, −1), (a − b, −b − 1), (2 + b + c − m, 1 + b) 2 Ψ3
10
X. Wang
Applying the contiguous relation (23) on the first Fox–Wright function in the right-hand side of the above identity, we just arrive at
(a, −b), (1 + c, b) 1 2 Ψ3 (m + 2, −1), (1 + a, −b − 1), (c − m, 1 + b)
1 (a, −b), (1 + c, b) = (c − m) Ψ 1 2 3 (m + 1, −1), (1 + a, −b − 1), (1 + c − m, 1 + b) m+1 1 (a − b, −b), (1 + b + c, b) + Ψ 1 2 3 (m + 1, −1), (a − b, −b − 1), (2 + b + c − m, 1 + b) m+1 (a − b, −b), (1 + b + c, b) +(1 + b)2 Ψ3 1 (m + 1, −1), (a − b, −b − 1), (2 + b + c − m, 1 + b) c−m (a, −b), (1 + c, b) = Ψ 1 2 3 (m + 1, −1), (1 + a, −b − 1), (1 + c − m, 1 + b) m+1 1 + b + c + bm (a − b, −b), (1 + b + c, b) + 1 . 2 Ψ3 (m + 1, −1), (a − b, −b − 1), (2 + b + c − m, 1 + b) m+1 It is obvious that the first 2 Ψ3 -series in the right-hand side of the above identity is just the identity (27) with n = m. Reversing the summation order of the second 2 Ψ3 -series in the right-hand side of the above identity, we have 2 Ψ3
=
(a − b, −b), (1 + b + c, b) 1 (m + 1, −1), (a − b, −b − 1), (2 + b + c − m, 1 + b)
m k=0
=
m k=0
= 2 Ψ3
Γ(a − b − bk)Γ(1 + b + c + bk) 1 Γ(m + 1 − k)Γ(a − b − bk − k)Γ(2 + b + c − m + bk + k) k! Γ(a − b − bm + bk)Γ(1 + b + c + bm − bk) 1 Γ(m + 1 − k)Γ(a − b − bm − m + bk + k)Γ(2 + b + c + bm − bk − k) k!
(a − b − bm, b), (1 + b + c + bm, −b) 1 , (m + 1, −1), (a − b − bm − m, b + 1), (2 + b + c + bm, −1 − b)
where we have performed the replacement k → m − k in the second equality. It is obvious that the 2 Ψ3 -series in the right-hand side of the above identity is just the Hagen-Rothe formula (27) with the replacement a → 1 + b + c + bm and c → a − b − bm − 1. Then we can calculate that
(a, −b), (1 + c, b) 1 (m + 2, −1), (1 + a, −b − 1), (c − m, 1 + b)
a+c 1 + b + c + bm a+c c−m 1 + = m m a(1 + m) 1+m 1 + b + c + bm
1 a+c = , a m+1 2 Ψ3
which is just the identity (27) with n = m + 1. This completes the proof of the Hagen–Rothe convolutions.
Contiguous relations for the Fox–Wright function
11
Acknowledgement This work is supported by National Natural Science Foundations of China (11661032). References [1] Andrews GE, Askey R, Roy R. Special Functions. Cambridge Univ. Press, Cambridge, 2000. [2] Brychkov YuA, Saad N. Some formulas for the Appell function F1 (a, b, b ; c; w, z). Integral Transforms and Special Functions. 2012; 23(11): 793–802. [3] Brychkov YuA, Saad N. On some formulas for the Appell function F2 (a, b, b ; c, c ; w, z). Integral Transforms and Special Functions. 2014; 25(2): 111–123. [4] Brychkov YuA, Saad N. On some formulas for the Appell function F3 (a, a , b, b ; c; w, z). Integral Transforms and Special Functions. 2015; 26(11): 910–923. [5] Brychkov YuA, Saad N. On some formulas for the Appell function F4 (a, b; c, c ; w, z). Integral Transforms and Special Functions. 2017; 28(9): 629–644. [6] Chu W, Wang X. Abel’s method on summation by parts and hypergeometric contiguous relations. Integral Transforms and Special Functions. 2007; 28(11): 771–807. [7] Chu W, Wang X. Summation formulae on Fox–Wright Ψ-functions. Integral Transforms and Special Functions. 2008; 19(8): 545–561. [8] Craven T, Csorads G. The Fox–Wright functions and Laguerre multiplier sequences. J. Math. Anal. Appl. 2006; 314: 109–125. [9] Darus M, Ibrahim RW. On Ces´ aro means for Fox–Wright functions. J. Math. Stat. 2008; 4: 156–160. [10] Gould HW. Some generalizations of Vandermonde’s convolution. Amer. Math. Month., 1956; 63: 84–91. [11] Fox C. The asymptotic expansion of generalized hypergeometric functions. Proc. Lond. Math. Soc. 1928; 27 (3) (Ser. 2): 389–400. [12] Krattenthaler C, Rivoal T. How can we escape Thomae’s relations? J. Math. Soc. Japan 2006; 58: 183–210. [13] Mainardi F, Pagnini G. The role of the Fox–Wright functions in fractional sub–diffusion of distributed order. J. Comput. Appl. Math. 2007; 207: 245–257. [14] Miller AR, Moskowitz IS. Reduction of a class of Fox–Wright Psi functions for certain rational parameters. Comput. Math. Appl. 1995; 30: 73–82. [15] Miller AR. On the Mellin transform of a product of two Fox–Wright psi functions. J. Phys. A: Math. Gen. 2002; 35: 2275–2281. [16] Opps SB, Saad N, Srivastava HM. Recursion formulas for Appell’s hypergeometric function F2 with some applications to radiation field problem. Applied Mathematics and Computation. 2009; 207: 545–558. [17] Srivastava HM, Karlsson PW. Multiple Gaussian Hypergeometric Series. Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York; 1985. [18] Wang X. Recursion formulas for Appell functions. Integral Transformations and Special Functions. 2012; 23(6): 421–433. [19] Wei C, Gong D, Hao H. A family of summation formulas on the Fox–Wright function. J. Math. Anal. Appl. 2012; 393: 316–327. [20] Whipple JF. A group of generalized hypergeometric series: relations between 120 allied series of type F[a, b, c; d, e]. Proceedings of the London Mathematical Society, 1925; 23(2): 104–C114.. [21] Wright EM. The asymptotic expansion of generalized hypergeometric functions. J. Lond. Math. Soc. 1935; 10: 286–293. [22] Wright EM. The asymptotic expansion of generalized hypergeometric functions. Proc. Lond. Math. Soc. 1940; 46 (3)(Ser. 2): 389–408.