Abstract
We give a simple proof of Slater’s transformations for bilateral series \({}_r\psi _r\), \({}_{2r}\psi _{2r}\), and \({}_{2r-1}\psi _{2r-1}\), using the residue theorem only, without technical manipulation of the series.
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1 Introduction
The bilateral basic hypergeometric series \({}_n\psi _n\) is defined by
for
where
and
In [6, 7], Slater derived transformations for bilateral basic hypergeometric series. She applied the results by Sears [5] in [6], and she considered the basic analogue of the Barnes-type integrals in [7]. Exposition on the work by Slater is provided in Chapter 5 of the book by Gasper and Rahman [1]. On the other hand, Ito and Sanada gave a proof of the transformation for \({}_r\psi _r\) series (Theorem 1 below) and a proof of the transformation for very-well-poised-balanced \({}_{2r}\psi _{2r}\) series (Theorem 2 below) from the viewpoint of the connection problem associated with a Jackson integral [2].
The purpose of the present paper is to give a simple proof of these results by means of the residue theorem only, without technical manipulation of the series, and also to study the integrals which represent several functions in q-analysis. We refer the reader to our previous work [4] for the proof of Ramanujan’s \({}_1\psi _1\)-sum and Bailey’s \({}_6\psi _6\)-sum from the same point of view (See also [3]).
We also use the symbols
and
In this paper, the base q is fixed to be a real number satisfying \(0<q<1\) for simplicity.
2 \({}_{r}\psi _{r}\) series
Slater’s transformation for \({}_{r}\psi _{r}\) series is given by the following, which is (5.4.3) of [1] ( (4) of [6], (7.2.5) of [7], and (5.1) of [2] ):
Theorem 1
Suppose that
for \(r\ge 1.\) Then we have
where \(A=\prod _{i=1}^ra_ic_i^{-1}\) and \(\left| \,b_1\cdots b_ra_1^{-1}\cdots a_r^{-1}\,\right|<|\,x\,|<1\).
Proof
Let F(t) be a function defined by
First, for \(k\in {\mathbb {Z}}\), we have
and
Secondly, for real positive numbers \(R_1\) and \(R_2\) satisfying
we have
and thus
where \(l\in {\mathbb {Z}}_{\ge 0},\) \(C_1^{(l)}= \{\,R_1\,e^{\sqrt{-1}\theta }q^{-l}\in {\mathbb {C}}\mid 0\le \theta \le 2\pi \}, C_2^{(l)}= \{\,R_2\,e^{\sqrt{-1}\theta }q^l\in {\mathbb {C}}\mid 0\le \theta \le 2\pi \},\) and \(M_1,\,M_2\) are positive numbers independent of l.
The residue theorem combined with inequalities (2.5) and (2.6) leads to
if \(\left| \,b_1\cdots b_r a_1^{-1}\cdots a_r^{-1}\,\right|<|\,x\,|<1,\) where \(C_1^{(l)}\) is in the counterclockwise direction and \(C_2^{(l)}\) is in the clockwise direction.
Consequently, combining (2.3), (2.4) with (2.7), we obtain
if \(\left| \,b_1\cdots b_r a_1^{-1}\cdots a_r^{-1}\,\right|<|\,x\,|<1.\) This completes the proof of (2.1). \(\square \)
3 Very-well-poised-balanced \({}_{2r}\psi _{2r}\) series
Slater’s transformation for very-well-poised-balanced \({}_{2r}\psi _{2r}\) series is given by the following, which is (5.5.2) of [1] ( (1.1) of [2] ):
Theorem 2
Suppose that
for \(r\ge 3\), and \(\left| \,a^{r-1}q^{r-2}\,\right| <\left| \,\prod _{i=1}^{2(r-1)}b_i\,\right| .\) Then we have
Proof
Let F(t) be a function defined by
which satisfies \(F(t)=F(a^{-1}t^{-1}).\)
First, for \(k\in {\mathbb {Z}}\), we have
and
Secondly, for \(k\in {\mathbb {Z}}\), we have
and
by the change of integration variable from t to u by \(t=a^{-1}u^{-1}\) with the equality \(F(t)=F(a^{-1}t^{-1}).\)
Thirdly, for real positive numbers \(R_1\) and \(R_2\) satisfying
we have
and thus
where \(l\in {\mathbb {Z}}_{\ge 0},\) \(C_1^{(l)}= \{\,R_1\,e^{\sqrt{-1}\theta }q^{-l}\in {\mathbb {C}}\mid 0\le \theta \le 2\pi \}, C_2^{(l)}= \{\,R_2\,e^{\sqrt{-1}\theta }q^l\in {\mathbb {C}}\mid 0\le \theta \le 2\pi \},\) and \(M_1,\,M_2\) are positive numbers independent of l.
The residue theorem combined with inequalities (3.7) and (3.8) leads to
if \(\left| \,a^{r-1}q^{r-2}\,\right| <\left| \,\prod _{i=1}^{2(r-1)}b_i\,\right| ,\) where \(C_1^{(l)}\) is in the counterclockwise direction and \(C_2^{(l)}\) is in the clockwise direction.
On the other hand, for \(k\in {\mathbb {Z}}\), we have
Consequently, combining (3.10), (3.11) with (3.9), we obtain
which is equivalent to
This completes the proof of (3.1). \(\square \)
Remark 1
If we change r to \(r+2\) and substitute \(b_{2r+1}=a^{\frac{1}{2}},\; b_{2r+2}=-a^{\frac{1}{2}}\) in (3.1), we obtain (5.5.1) of [1] ((7) of [6], and (7.2.1.1) of [7]), since
Remark 2
To obtain (5.5.1) of [1] by the same manner as in the proof of Theorem 2, it is enough to consider the function
4 Very-well-poised \({}_{2r}\psi _{2r}\) and \({}_{2r-1}\psi _{2r-1}\) series
If we change r to \(r+1\) and substitute \(b_{2r-1}=q^{\frac{1}{2}}a^{\frac{1}{2}},\; b_{2r}=-q^{\frac{1}{2}}a^{\frac{1}{2}}\) in (3.1), we obtain the following transformation for very-well-poised \({}_{2r}\psi _{2r}\) series:
Theorem 3
Suppose that
for \(r\ge 2\), and \(\left| \,a^{r-2}q^{r-1}\,\right| <\left| \,\prod _{i=1}^{2(r-1)}b_i\,\right| .\) Then we have
Remark 3
To obtain (4.1) by the same manner as in the proof of Theorem 2, it is enough to consider the function
Remark 4
If we change r to \(r+2\) and substitute \(b_{2r+1}=a^{\frac{1}{2}},\; b_{2r+2}=-a^{\frac{1}{2}}\) in (4.1), we obtain (5.5.4) of [1] ( (8) of [6], and (7.2.1.3) of [7] ), since
and
Remark 5
To obtain (5.5.4) of [1] by the same manner as in the proof of Theorem 2, it is enough to consider the function
The substitution \(b_{2(r-1)}=\pm q^{\frac{1}{2}}a^{\frac{1}{2}}\) in (4.1) implies the following transformation for very-well-poised \({}_{2r-1}\psi _{2r-1}\) series:
Theorem 4
Suppose that
for \(r\ge 3\), and \(\left| \,a^{r-\frac{5}{2}}q^{r-\frac{3}{2}}\,\right| <\left| \,\prod _{i=1}^{2r-3}b_i\,\right| .\) Then we have
Here either all the upper or all the lower signs are taken throughout.
Remark 6
To obtain (4.2) by the same manner as in the proof of Theorem 2, it is enough to consider the function
Remark 7
If we change r with \(r+2\) and substitute \(b_{2r}=a^{\frac{1}{2}},\; b_{2r+1}=-a^{\frac{1}{2}}\) in (4.2), we obtain (5.5.5) of [1] ( (9) of [6], and (7.2.1.4) of [7]), since
and
Remark 8
To obtain (5.5.5) of [1] by the same manner as in the proof of Theorem 2, it is enough to consider the function
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References
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Mimachi, K. A simple proof of Slater’s transformations for bilateral series \({}_{r}\psi _{r},\, {}_{2r}\psi _{2r}\) and \({}_{2r-1}\psi _{2r-1}\). Ramanujan J 64, 67–78 (2024). https://doi.org/10.1007/s11139-023-00812-3
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DOI: https://doi.org/10.1007/s11139-023-00812-3
Keywords
- Slater’s transformations
- \({}_{r}\psi _{r}\)
- Very-well-poised \({}_{2r}\psi _{2r}\) series
- Bilateral basic hypergeometric series