SOME MORE SEMI-FINITE FORMS OF BILATERAL BASIC HYPERGEOMETRIC SERIES

FRÉDÉRIC JOUHET^{∗}

Abstract. We prove some new semi-finite forms of bilateral basic hypergeometric series. One of them yields in a direct limit Bailey’s celebrated _{6}ψ_{6} summation formula, answering a question recently raised by Chen and Fu (Semi-Finite Forms of Bilateral Basic Hypergeometric Series, Proc. Amer. Math. Soc., to appear).

1. Introduction

There is a standard method for obtaining a bilateral identity from a unilateral terminating identity, which was already utilized by Cauchy [7] in his second proof of Jacobi’s [10] famous triple product identity. The idea of this method is to start from a finite unilateral summation and to shift the index of summation, say k (0 ≤ k ≤ 2n), by n :

∑_{k=0}^{2n} a(k) = ∑_{k=−n}^{n} a(k + n),  (1.1)

and then let n → ∞ whenever it is possible after some manipulations. The same method has also been exploited by Bailey [4, Secs. 3 and 6], [5], Slater [14, Sec. 6.2], Schlosser [13] and Schlosser and the author [11]. Recently, Chen and Fu [8] used a method different from the previous one, as they started from unilateral infinite summations to derive semi-finite forms of bilateral basic hypergeometric series. The process can be summarized as follows :

∑_{k≥0} a(k) = ∑_{k≥−n} a(k + n),  (1.2)

and then let n → ∞ whenever it is possible after some manipulations. The righthand side of (1.1) (resp. (1.2)) can be seen as a finite (resp. semi-finite) form of a bilateral series. Chen and Fu have found in [8] semi-finite forms of Ramanujan’s 1ψ_{1} summation formula (cf. [9, Appendix (II.29)]), of a _{2}ψ_{2} formula due to Bailey [9, Ex. 5.20(i)], and of Bailey’s [4, Eq. (4.7)] _{6}ψ_{6} summation formula ([9, Appendix (II.33)]), which can be written as follows :

6ψ_{6}[\begin{matrix}q\sqrt{a}, −q\sqrt{a}, b, c, d, e \\ \sqrt{a}, −\sqrt{a}, aq/b, aq/c, aq/d, aq/e \end{matrix}; q, \frac{qa^{2}}{bcde}]

= \frac{(q, aq, q/a, aq/bc, aq/bd, aq/be, aq/cd, aq/ce, aq/de)_{∞}}{(q/b, q/c, q/d, q/e, aq/b, aq/c, aq/d, aq/e, qa^{2}/bcde)_{∞}} (1.3)

2000 Mathematics Subject Classification. 33D15.

Key words and phrases. bilateral basic hypergeometric series, q-series, Bailey’s_{6}ψ_{6} summation.

∗Partially supported by EC’s IHRP Programme, grant HPRN-CT-2001-00272, “Algebraic Combinatorics in Europe”.

(see the end of this introduction for the notations), where |q| < 1 and |qa^{2}/bcde| < 1. At the end of [8], Chen and Fu mention the problem of finding a proof of (1.3) using a semi-finite (or even finite) form which would yield (1.3) in a direct limit. Indeed, after letting n → ∞ in their semi-finite form of (1.3), one needs to use Ramanujan’s 1ψ_{1} summation formula to derive (1.3). In this paper, we use the method developped in [8] to find, among other results, a new semi-finite form of (1.3) which answers the question raised by Chen and Fu. After explaining some notations in the end of this introduction, we show in section 2 how the method in [8] can be applied to yield in a direct limit (1.3), starting from a nonterminating extension of Jackson’s formula due to Bailey [9, Appendix (II.25)]. We give two other applications of this method in section 3, which yield in a direct limit a transformation formula for a _{6}ψ_{6} series proved in [11] and a transformation formula for a _{8}ψ_{8} series in terms of two _{8}ϕ_{7} series and a _{8}ψ_{8} series.

Other proofs of Bailey’s very-well-poised _{6}ψ_{6} summation had been given by Bailey [4], Slater and Lakin [15], Andrews [1], Askey and Ismail [3], Askey [2], Schlosser [12] and Schlosser and the author [11]. It is worth noting that the elegant proof of Askey and Ismail in [3] uses an argument of analytic continuation together with the shift (1.2), but used from right to left.

Notation: It is appropriate to recall some standard notations for q-series and basic hypergeometric series.

Let q be a fixed complex parameter (the “base”) with 0 < |q| < 1. The q-shifted factorial is defined for any complex parameter a by

(a)_{∞} ≡ (a; q)_{∞} := ∏_{j≥0}(1 − aq^{j})

and

(a)_{k} ≡ (a; q)_{k} := \frac{(a; q)_{∞}}{(aq^{k}; q)_{∞}} ,

where k is any integer. Since the same base q is used throughout this paper, it may be readily omitted (in notation, writing (a)_{k} instead of (a; q)_{k}, etc) which will not lead to any confusion. For brevity, write

(a_{1}, …, a_{m})_{k} := (a_{1})_{k} ⋯ (a_{m})_{k},

where k is an integer or infinity. Further, recall the definition of basic hypergeometric series,

sϕ_{s−1}[\begin{matrix}a_{1}, …, a_{s} \\ b_{1}, …, b_{s−1} \end{matrix}; q, z] := ∑_{k=0}^{∞} \frac{(a_{1}, …, a_{s})_{k}}{(q, b_{1}, …, b_{s−1})_{k}} z^{k},

and of bilateral basic hypergeometric series,

sψ_{s}[\begin{matrix}a_{1}, …, a_{s} \\ b_{1}, …, b_{s} \end{matrix}; q, z] := ∑_{k=−∞}^{∞} \frac{(a_{1}, …, a_{s})_{k}}{(b_{1}, …, b_{s})_{k}} z^{k}.

See Gasper and Rahman’s text [9] for a comprehensive study of the theory of basic hypergeometric series. In particular, the computations in this paper rely on some elementary identities for q-shifted factorials, listed in [9, Appendix I].

2. A new semi-finite form of Bailey’s _{6}ψ_{6} summation formula

Consider Bailey’s nonterminating extension of Jackson’s _{8}ϕ_{7} summation [9, Appendix (II.25)]

8ϕ_{7}[\begin{matrix}a, q\sqrt{a}, −q\sqrt{a}, b, c, d, e, f \\ \sqrt{a}, −\sqrt{a}, aq/b, aq/c, aq/d, aq/e, aq/f \end{matrix}; q, q]

= \frac{b}{a} \frac{(aq, c, d, e, f, bq/a, bq/c, bq/d, bq/e, bq/f)_{∞}}{(aq/b, aq/c, aq/d, aq/e, aq/f, bc/a, bd/a, be/a, bf/a, b^{2}q/a)_{∞}}

×_{8}ϕ_{7}[\begin{matrix}b^{2}/a, qb/\sqrt{a}, −qb/\sqrt{a}, b, bc/a, bd/a, be/a, bf/a \\ b/\sqrt{a}, −b/\sqrt{a}, bq/a, bq/c, bq/d, bq/e, bq/f \end{matrix}; q, q]

+ \frac{(aq, b/a, aq/cd, aq/ce, aq/cf, aq/de, aq/df, aq/ef)_{∞}}{(aq/c, aq/d, aq/e, aq/f, bc/a, bd/a, be/a, bf/a)_{∞}} , (2.1)

where qa^{2} = bcdef. Note that (2.1) can be proved by specializing qa^{2} = bcdef in Bailey’s 3-term transformation formula for a nonterminating very-well-poised _{8}ϕ_{7} [9, Appendix (III.37)], which was the starting point in [8] to prove (1.3), and then using the sum of a verywell-poised _{6}ϕ_{5} [9, Appendix (II.20)]. Now, using (2.1), we can derive the following semi-finite form of (1.3).

Proposition 2.1.

∑_{k≥−n} \frac{(aq^{−n}, q\sqrt{a}, −q\sqrt{a}, bq^{n}, c, d, e, f)_{k}}{(q^{1+n}, \sqrt{a}, −\sqrt{a}, aq^{1−n}/b, aq/c, aq/d, aq/e, aq/f)_{k}} q^{k}

= \frac{(aq, c, d, e, f, bq^{1+2n}/a, bq^{1+n}/c, bq^{1+n}/d, bq^{1+n}/e, bq^{1+n}/f)_{∞}}{(aq/b, aq/c, aq/d, aq/e, aq/f, bcq^{n}/a, bdq^{n}/a, beq^{n}/a, bfq^{n}/a, b^{2}q^{1+2n}/a)_{∞}}

× \frac{b^{n+1}}{a} \frac{(q, q/a)_{n}}{(b, b/a)_{n}}

×_{8}ϕ_{7}[\begin{matrix}b^{2}q^{2n}/a, bq^{1+n}/\sqrt{a}, −bq^{1+n}/\sqrt{a}, b, bcq^{n}/a, bdq^{n}/a, beq^{n}/a, bfq^{n}/a \\ bq^{n}/\sqrt{a}, −bq^{n}/\sqrt{a}, bq^{1+2n}/a, bq^{1+n}/c, bq^{1+n}/d, bq^{1+n}/e, bq^{1+n}/f \end{matrix}; q, q]

+ \frac{(aq, aq/cd, aq/ce, aq/cf, aq/de, aq/df, aq/ef, bq^{n}/a)_{∞}}{(aq/c, aq/d, aq/e, aq/f, bcq^{n}/a, bdq^{n}/a, beq^{n}/a, bfq^{n}/a)_{∞}}

× \frac{(q, q/a)_{n}}{(b, q/c, q/d, q/e, q/f)_{n}} , (2.2)

where b = qa^{2}/cdef.

Proof. By shifting the index of summation by n, the left-hand side of (2.1) is equal to

\frac{1 − aq^{2n}}{1 − a} \frac{(a, b, c, d, e, f)_{n}}{(q, aq/b, aq/c, aq/d, aq/e, aq/f)_{n}} q^{n}

× ∑_{k≥−n} \frac{(aq^{n}, \sqrt{a}q^{1+n}, −\sqrt{a}q^{1+n}, bq^{n}, cq^{n}, dq^{n}, eq^{n}, fq^{n})_{k}}{(q^{1+n}, \sqrt{a}q^{n}, −\sqrt{a}q^{n}, aq/b, aq^{1+n}/c, aq^{1+n}/d, aq^{1+n}/e, aq^{1+n}/f)_{k}} q^{k}.

Next, on both sides of (2.1), replace a, c, d, e and f by aq^{−2n}, cq^{−n}, dq^{−n}, eq^{−n} and fq^{−n} respectively. Note that the condition qa^{2} = bcdef is equivalent to b = qa^{2}/cdef, thus b remains unchanged. We get

\frac{1 − a}{1 − aq^{−2n}} \frac{(aq^{−2n}, b, cq^{−n}, dq^{−n}, eq^{−n}, fq^{−n})_{n}}{(q, aq^{1−2n}/b, aq^{1−n}/c, aq^{1−n}/d, aq^{1−n}/e, aq^{1−n}/f)_{n}} q^{n}

× ∑_{k≥−n} \frac{(aq^{−n}, q\sqrt{a}, −q\sqrt{a}, bq^{n}, c, d, e, f)_{k}}{(q^{1+n}, \sqrt{a}, −\sqrt{a}, aq^{1−n}/b, aq/c, aq/d, aq/e, aq/f)_{k}} q^{k}

= \frac{b}{aq^{−2n}} \frac{(aq^{1−2n}, cq^{−n}, dq^{−n}, eq^{−n}, fq^{−n})_{∞}}{(aq^{1−2n}/b, aq^{1−n}/c, aq^{1−n}/d, aq^{1−n}/e, aq^{1−n}/f)_{∞}}

× \frac{(bq^{1+2n}/a, bq^{1+n}/c, bq^{1+n}/d, bq^{1+n}/e, bq^{1+n}/f)_{∞}}{(bcq^{n}/a, bdq^{n}/a, beq^{n}/a, bfq^{n}/a, b^{2}q^{1+2n}/a)_{∞}}

×_{8}ϕ_{7}[\begin{matrix}b^{2}q^{2n}/a, bq^{1+n}/\sqrt{a}, −bq^{1+n}/\sqrt{a}, b, bcq^{n}/a, bdq^{n}/a, beq^{n}/a, bfq^{n}/a \\ bq^{n}/\sqrt{a}, −bq^{n}/\sqrt{a}, bq^{1+2n}/a, bq^{1+n}/c, bq^{1+n}/d, bq^{1+n}/e, bq^{1+n}/f \end{matrix}; q, q]

+ \frac{(aq^{1−2n}, bq^{2n}/a, aq/cd, aq/ce, aq/cf, aq/de, aq/df, aq/ef)_{∞}}{(aq^{1−n}/c, aq^{1−n}/d, aq^{1−n}/e, aq^{1−n}/f, bcq^{n}/a, bdq^{n}/a, beq^{n}/a, bfq^{n}/a)_{∞}} .

This can be rewritten as

∑_{k≥−n} \frac{(aq^{−n}, q\sqrt{a}, −q\sqrt{a}, bq^{n}, c, d, e, f)_{k}}{(q^{1+n}, \sqrt{a}, −\sqrt{a}, aq^{1−n}/b, aq/c, aq/d, aq/e, aq/f)_{k}} q^{k}

= \frac{1 − aq^{−2n}}{1 − a} \frac{bq^{n}}{a} \frac{(q, aq^{1−2n}/b)_{n}}{(b, aq^{−2n})_{n}} \frac{(aq^{1−2n})_{∞}}{(aq^{1−2n}/b)_{∞}}

× \frac{(c, d, e, f, bq^{1+2n}/a, bq^{1+n}/c, bq^{1+n}/d, bq^{1+n}/e, bq^{1+n}/f)_{∞}}{(aq/c, aq/d, aq/e, aq/f, bcq^{n}/a, bdq^{n}/a, beq^{n}/a, bfq^{n}/a, b^{2}q^{1+2n}/a)_{∞}}

×_{8}ϕ_{7}[\begin{matrix}b^{2}q^{2n}/a, bq^{1+n}/\sqrt{a}, −bq^{1+n}/\sqrt{a}, b, bcq^{n}/a, bdq^{n}/a, beq^{n}/a, bfq^{n}/a \\ bq^{n}/\sqrt{a}, −bq^{n}/\sqrt{a}, bq^{1+2n}/a, bq^{1+n}/c, bq^{1+n}/d, bq^{1+n}/e, bq^{1+n}/f \end{matrix}; q, q]

+ \frac{1 − aq^{−2n}}{1 − a} q^{−n} \frac{(q, aq^{1−2n}/b)_{n}(aq^{1−2n})_{∞}}{(aq^{−2n}, b, cq^{−n}, dq^{−n}, eq^{−n}, fq^{−n})_{n}}

× \frac{(bq^{2n}/a, aq/cd, aq/ce, aq/cf, aq/de, aq/df, aq/ef)_{∞}}{(aq/c, aq/d, aq/e, aq/f, bcq^{n}/a, bdq^{n}/a, beq^{n}/a, bfq^{n}/a)_{∞}} . (2.3)

Now we use the three following elementary identities to simplify the right-hand side of (2.3) :

\frac{(xq^{−2n})_{∞}}{(xq^{−2n})_{n}} = (−1)^{n}x^{n}q^{−(n^{2}+n)/2}(q/x)_{n}(x)_{∞},  (2.4)
(xq^{−2n})_{n} = (−1)^{n}x^{n}q^{−(3n^{2}+n)/2}(q^{n+1}/x)_{n},  (2.5)
(xq^{−n})_{n} = (−1)^{n}x^{n}q^{−(n^{2}+n)/2}(q/x)_{n},  (2.6)

and we obtain (2.2) after simplifications. □

Now, one may let n → ∞ in (2.2), assuming |qa^{2}/cdef| < 1 (i.e. |b| < 1), while appealing to Tannery’s theorem [6] for being able to interchange limit and summation. As the first term on the right-hand side of (2.2) tends to 0, this gives immediately Bailey’s _{6}ψ_{6} summation formula (1.3) with b replaced by f.

3. Other consequences

We give in this section two other applications of the previous process. Consider first the following transformation formula for a non terminating very-well-poised 8ϕ_{7} series [9, Appendix (III.23)]

8ϕ_{7}[\begin{matrix}a, q\sqrt{a}, −q\sqrt{a}, b, c, d, e, f \\ \sqrt{a}, −\sqrt{a}, aq/b, aq/c, aq/d, aq/e, aq/f \end{matrix}; q, \frac{q^{2}a^{2}}{bcdef} ]

= \frac{(aq, aq/ef, λq/e, λq/f)_{∞}}{(aq/e, aq/f, λq/ef, λq)_{∞}}

×_{8}ϕ_{7}[\begin{matrix}λ, q\sqrt{λ}, −q\sqrt{λ}, λb/a, λc/a, λd/a, e, f \\ \sqrt{λ}, −\sqrt{λ}, aq/b, aq/c, aq/d, λq/e, λq/f \end{matrix}; q, \frac{aq}{ef} ] , (3.1)

where λ = qa^{2}/bcd, |q^{2}a^{2}/bcdef| < 1 and |aq/ef| < 1. Note that (3.1) is nothing else but the n → ∞ case of Bailey’s [4] transformation formula for a very-well-poised _{10}ϕ_{9} series [9, Appendix (III.28)], which was the starting point in [11] for the derivation of (1.3). Now, using (3.1), we can prove the following semi-finite identity.

Proposition 3.1.

∑_{k≥−n} \frac{(aq^{−n}, q\sqrt{a}, −q\sqrt{a}, bq^{n}, c, d, e, f)_{k}}{(q^{1+n}, \sqrt{a}, −\sqrt{a}, aq^{1−n}/b, aq/c, aq/d, aq/e, aq/f)_{k}} (\frac{q^{2}a^{2}}{bcdef} )^{k}

= \frac{(aq, aq/ef, λq/e, λq/f)_{∞}}{(aq/e, aq/f, λq/ef, λq)_{∞}} \frac{(λb/a, q/a, aq/λc, aq/λd)_{n}}{(b, q/λ, q/c, q/d)_{n}}

× ∑_{k≥−n} \frac{(λq^{−n}, q\sqrt{λ}, −q\sqrt{λ}, λbq^{n}/a, λc/a, λd/a, e, f)_{k}}{(q^{1+n}, \sqrt{λ}, −\sqrt{λ}, aq^{1−n}/b, aq/c, aq/d, λq/e, λq/f)_{k}} (\frac{aq}{ef} )^{k} , (3.2)

where λ = qa^{2}/bcd and |q^{2}a^{2}/bcdef| < 1.

Proof. By shifting the index of summation by n on both sides of (3.1), we get

\frac{1 − aq^{2n}}{1 − a} \frac{(a, b, c, d, e, f)_{n}}{(q, aq/b, aq/c, aq/d, aq/e, aq/f)_{n}} (\frac{q^{2}a^{2}}{bcdef} )^{n}

× ∑_{k≥−n} \frac{(aq^{n}, \sqrt{a}q^{1+n}, −\sqrt{a}q^{1+n}, bq^{n}, cq^{n}, dq^{n})_{k}}{(q^{1+n}, \sqrt{a}q^{n}, −\sqrt{a}q^{n}, aq^{1+n}/b, aq^{1+n}/c, aq^{1+n}/d)_{k}}

× \frac{(eq^{n}, fq^{n})_{k}}{(aq^{1+n}/e, aq^{1+n}/f)_{k}} (\frac{q^{2}a^{2}}{bcdef} )^{k}

= \frac{(aq, aq/ef, λq/e, λq/f)_{∞}}{(aq/e, aq/f, λq/ef, λq)_{∞}} \frac{1 − λq^{2n}}{1 − λ} \frac{(λ, λb/a, λc/a, λd/a, e, f)_{n}}{(q, aq/b, aq/c, aq/d, λq/e, λq/f)_{n}} (\frac{aq}{ef} )^{n}

× ∑_{k≥−n} \frac{(λq^{n}, \sqrt{λ}q^{1+n}, −\sqrt{λ}q^{1+n}, λbq^{n}/a, λcq^{n}/a, λdq^{n}/a)_{k}}{(q^{1+n}, \sqrt{λ}q^{n}, −\sqrt{λ}q^{n}, aq^{1+n}/b, aq^{1+n}/c, aq^{1+n}/d)_{k}}

× \frac{(eq^{n}, fq^{n})_{k}}{(λq^{1+n}/e, λq^{1+n}/f)_{k}} (\frac{aq}{ef} )^{k} . (3.3)

Next, on both sides of (3.3), replace a, c, d, e and f by aq^{−2n}, cq^{−n}, dq^{−n}, eq^{−n} and fq^{−n} respectively. Note that the condition λ = qa^{2}/bcd implies that λ is replaced by λq^{−2n}. This yields

∑_{k≥−n} \frac{(aq^{−n}, q\sqrt{a}, −q\sqrt{a}, bq^{n}, c, d, e, f)_{k}}{(q^{1+n}, \sqrt{a}, −\sqrt{a}, aq^{1−n}/b, aq/c, aq/d, aq/e, aq/f)_{k}} (\frac{q^{2}a^{2}}{bcdef} )^{k}

= \frac{1 − aq^{−2n}}{1 − a} \frac{1 − λ}{1 − λq^{−2n}} \frac{(λq^{−2n}, λb/a, λcq^{−n}/a, λdq^{−n}/a)_{n}}{(aq^{−2n}, b, cq^{−n}, dq^{−n})_{n}} (\frac{a}{λ})^{n}

× \frac{(aq^{1−2n}, aq/ef, λq/e, λq/f)_{∞}}{(λq^{1−2n}, aq/e, aq/f, λq/ef)_{∞}}

× ∑_{k≥−n} \frac{(λq^{−n}, q\sqrt{λ}, −q\sqrt{λ}, λbq^{n}/a, λc/a, λd/a, e, f)_{k}}{(q^{1+n}, \sqrt{λ}, −\sqrt{λ}, aq^{1−n}/b, aq/c, aq/d, λq/e, λq/f)_{k}} (\frac{aq}{ef} )^{k} ,

which is (3.2) after using the simplifications (2.4) and (2.6) on the right-hand side. □

By letting n → ∞ in (3.2), assuming |qa^{2}/cdef| < 1 while appealing to Tannery’s theorem [6] for being able to interchange limit and summation, one gets the following transformation formula, which was derived in [11]

6ψ_{6}[\begin{matrix}q\sqrt{a}, −q\sqrt{a}, c, d, e, f \\ \sqrt{a}, −\sqrt{a}, aq/c, aq/d, aq/e, aq/f \end{matrix}; q, \frac{qa^{2}}{cdef} ]

= \frac{(aq, q/a, aq/ef, aq/cd, λq/e, λq/f, aq/λc, aq/λd)_{∞}}{(aq/e, aq/f, q/c, q/d, λq, q/λ, λq/ef, b)_{∞}}

×_{6}ψ_{6}[\begin{matrix}q\sqrt{λ}, −q\sqrt{λ}, λc/a, λd/a, e, f \\ \sqrt{λ}, −\sqrt{λ}, aq/c, aq/d, λq/e, λq/f \end{matrix}; q, \frac{qa^{2}}{cdef} ] , (3.4)

where λ = qa^{2}/bcd, and b is now an extra parameter on the right-hand side. As explained in [11], an iteration of (3.4) and an appropriate specialization of both extra parameters appearing on the right-hand side immediately establishes Bailey’s formula (1.3).

Now, we consider the next level in the hierarchy of identities for very-well-poised nonterminating basic hypergeometric series, which is Bailey’s four-term _{10}ϕ_{9} transformation [9, Appendix (III.39)]

10ϕ_{9}[\begin{matrix}a, q\sqrt{a}, −q\sqrt{a}, b, c, d, e, f, g, h \\ \sqrt{a}, −\sqrt{a}, aq/b, aq/c, aq/d, aq/e, aq/f, aq/g, aq/h \end{matrix}; q, q]

+\frac{(aq, b/a, c, d, e, f, g, h, bq/c, bq/d, bq/e, bq/f, bq/g, bq/h)_{∞}}{(b^{2}q/a, a/b, aq/c, aq/d, aq/e, aq/f, aq/g, aq/h, bc/a, bd/a, be/a, bf/a, bg/a, bh/a)_{∞}}

×_{10}ϕ_{9}[\begin{matrix}b^{2}/a, qb/\sqrt{a}, −qb/\sqrt{a}, b, bc/a, bd/a, be/a, bf/a, bg/a, bh/a \\ b/\sqrt{a}, −b/\sqrt{a}, bq/a, bq/c, bq/d, bq/e, bq/f, bq/g, bq/h \end{matrix}; q, q]

= \frac{(aq, b/a, λq/f, λq/g, λq/h, bf/λ, bg/λ, bh/λ)_{∞}}{(λq, b/λ, aq/f, aq/g, aq/h, bf/a, bg/a, bh/a)_{∞}}

×_{10}ϕ_{9}[\begin{matrix}λ, q\sqrt{λ}, −q\sqrt{λ}, b, λc/a, λd/a, λe/a, f, g, h \\ \sqrt{λ}, −\sqrt{λ}, λq/b, aq/c, aq/d, aq/e, λq/f, λq/g, λq/h \end{matrix}; q, q]

+\frac{(aq, b/a, f, g, h, bq/f, bq/g, bq/h, λc/a, λd/a, λe/a, abq/λc, abq/λd, abq/λe)_{∞}}{(b^{2}q/λ, λ/b, aq/c, aq/d, aq/e, aq/f, aq/g, aq/h, bc/a, bd/a, be/a, bf/a, bg/a, bh/a)_{∞}}

×_{10}ϕ_{9}[\begin{matrix}b^{2}/λ, qb/\sqrt{λ}, −qb/\sqrt{λ}, b, bc/a, bd/a, be/a, bf/λ, bg/λ, bh/λ \\ b/\sqrt{λ}, −b/\sqrt{λ}, bq/λ, abq/λc, abq/λd, abq/λe, bq/f, bq/g, bq/h \end{matrix}; q, q] , (3.5)

where λ = qa^{2}/cde and q^{2}a^{3} = bcdefgh. We can deduce from (3.5) the following semi-finite identity.

Proposition 3.2.

∑_{k≥−n} \frac{(aq^{−n}, q\sqrt{a}, −q\sqrt{a}, b, cq^{n}, d, e, f, g, h)_{k}}{(q^{1+n}, \sqrt{a}, −\sqrt{a}, aq/b, aq^{1−n}/c, aq/d, aq/e, aq/f, aq/g, aq/h)_{k}} q^{k}

+ α_{n} _{10}ϕ_{9}[\begin{matrix}b^{2}/a, bq/\sqrt{a}, −bq/\sqrt{a}, bq^{−n}, bcq^{n}/a, bd/a, be/a, bf/a, bg/a, bh/a \\ b/\sqrt{a}, −b/\sqrt{a}, bq^{1+n}/a, bq^{1−n}/c, bq/d, bq/e, bq/f, bq/g, bq/h \end{matrix}; q, q]

= β_{n} ∑_{k≥−n} \frac{(λq^{−n}, q\sqrt{λ}, −q\sqrt{λ}, b, λcq^{n}/a, λd/a, λe/a, f, g, h)_{k}}{(q^{1+n}, \sqrt{λ}, −\sqrt{λ}, λq/b, aq^{1−n}/c, aq/d, aq/e, λq/f, λq/g, λq/h)_{k}} q^{k}

+γ_{n} _{10}ϕ_{9}[\begin{matrix}b^{2}/λ, bq/\sqrt{λ}, −bq/\sqrt{λ}, bq^{−n}, bcq^{n}/a, bd/a, be/a, bf/λ, bg/λ, bh/λ \\ b/\sqrt{λ}, −b/\sqrt{λ}, bq^{1+n}/λ, abq^{1−n}/λc, abq/λd, abq/λe, bq/f, bq/g, bq/h \end{matrix}; q, q] , (3.6)

where λ = qa^{2}/cde, c = q^{2}a^{3}/bdefgh, and

α_{n} = −\frac{b}{a} \frac{(q, q/a, c/b)_{n}}{(q/b, c/a)_{n}} \frac{(bq^{1+n}/a, cq^{n}, aq, bq/c)_{∞}}{(bcq^{n}/a, b^{2}q/a, aq/b, aq/c)_{∞}}

× \frac{(bq/d, bq/e, bq/f, bq/g, bq/h, d, e, f, g, h)_{∞}}{(bd/a, be/a, bf/a, bg/a, bh/a, aq/d, aq/e, aq/f, aq/g, aq/h)_{∞}} ,

β_{n} = \frac{(q/a, λc/a, aq/λd, aq/λe, b/a)_{n}}{(q/λ, c, q/d, q/e, b/λ)_{n}}

× \frac{(aq, bf/λ, bg/λ, bh/λ, λq/f, λq/g, λq/h, bq^{n}/a)_{∞}}{(λq, bf/a, bg/a, bh/a, aq/f, aq/g, aq/h, bq^{n}/λ)_{∞}} ,

γ_{n} = \frac{(q, q/a, b/a, aq/λd, aq/λe, λc/ab)_{n}}{(c, c/a, q/b, q/d, q/e, qb/λ)_{n}} \frac{(bq^{n}/a, aq, f, g, h)_{∞}}{(bcq^{n}/a, b^{2}q/λ, aq/f, aq/g, aq/h)_{∞}}

× \frac{(λc/a, λd/a, λe/a, bq/f, bq/g, bq/h, abq/λc, abq/λd, abq/λe)_{∞}}{(bd/a, be/a, bf/a, bg/a, bh/a, λ/b, aq/c, aq/d, aq/e)_{∞}} .

Proof. In the first and the third summations of (3.5), shift the index of summation k by n, and replace a, b, d, e, f, g and h by aq^{−2n}, bq^{−n}, dq^{−n}, eq^{−n}, fq^{−n}, gq^{−n} and hq^{−n} respectively. Note that the condition λ = qa^{2}/cde implies that λ is replaced by λq^{−2n}, and the condition c = q^{2}a^{3}/bdefgh leaves c unchanged. The first _{10}ϕ_{9} is then equal to

δ_{n} ∑_{k≥−n} \frac{(aq^{−n}, q\sqrt{a}, −q\sqrt{a}, b, cq^{n}, d, e, f, g, h)_{k}}{(q^{1+n}, \sqrt{a}, −\sqrt{a}, aq/b, aq^{1−n}/c, aq/d, aq/e, aq/f, aq/g, aq/h)_{k}} q^{k},

where

δ_{n} = \frac{(aq^{−2n}, bq^{−n}, c, dq^{−n}, eq^{−n}, fq^{−n}, gq^{−n}, hq^{−n})_{n}}{(q, aq^{1−n}/b, aq^{1−2n}/c, aq^{1−n}/d, aq^{1−n}/e, aq^{1−n}/f, aq^{1−n}/g, aq^{1−n}/h)_{n}}

× \frac{1 − a}{1 − aq^{−2n}} q^{n},

and (3.5) is then equivalent to

∑_{k≥−n} \frac{(aq^{−n}, q\sqrt{a}, −q\sqrt{a}, b, cq^{n}, d, e, f, g, h)_{k}}{(q^{1+n}, \sqrt{a}, −\sqrt{a}, aq/b, aq^{1−n}/c, aq/d, aq/e, aq/f, aq/g, aq/h)_{k}} q^{k}

+ \frac{a_{n}}{δ_{n}} _{10}ϕ_{9}[\begin{matrix}b^{2}/a, bq/\sqrt{a}, −bq/\sqrt{a}, bq^{−n}, bcq^{n}/a, bd/a, be/a, bf/a, bg/a, bh/a \\ b/\sqrt{a}, −b/\sqrt{a}, bq^{1+n}/a, bq^{1−n}/c, bq/d, bq/e, bq/f, bq/g, bq/h \end{matrix}; q, q]

= \frac{b_{n}}{δ_{n}} ∑_{k≥−n} \frac{(λq^{−n}, q\sqrt{λ}, −q\sqrt{λ}, b, λcq^{n}/a, λd/a, λe/a, f, g, h)_{k}}{(q^{1+n}, \sqrt{λ}, −\sqrt{λ}, λq/b, aq^{1−n}/c, aq/d, aq/e, λq/f, λq/g, λq/h)_{k}} q^{k}

+\frac{c_{n}}{δ_{n}} _{10}ϕ_{9}[\begin{matrix}b^{2}/λ, bq/\sqrt{λ}, −bq/\sqrt{λ}, bq^{−n}, bcq^{n}/a, bd/a, be/a, bf/λ, bg/λ, bh/λ \\ b/\sqrt{λ}, −b/\sqrt{λ}, bq^{1+n}/λ, abq^{1−n}/λc, abq/λd, abq/λe, bq/f, bq/g, bq/h \end{matrix}; q, q] ,

where

a_{n} = \frac{(aq^{1−2n}, bq^{n}/a, c, dq^{−n}, eq^{−n}, fq^{−n}, gq^{−n}, hq^{−n})_{∞}}{(b^{2}q/a, aq^{−n}/b, aq^{1−2n}/c, aq^{1−n}/d, aq^{1−n}/e, aq^{1−n}/f, aq^{1−n}/g, aq^{1−n}/h)_{∞}}

× \frac{(bq^{1−n}/c, bq/d, bq/e, bq/f, bq/g, bq/h)_{∞}}{(bcq^{n}/a, bd/a, be/a, bf/a, bg/a, bh/a)_{∞}} ,

b_{n} = \frac{(aq^{1−2n}, bq^{n}/a, λq^{1−n}/f, λq^{1−n}/g, λq^{1−n}/h, bf/λ, bg/λ, bh/λ)_{∞}}{(λq^{1−2n}, bq^{n}/λ, aq^{1−n}/f, aq^{1−n}/g, aq^{1−n}/h, bf/a, bg/a, bh/a)_{∞}} \frac{1 − λ}{1 − λq^{−2n}} q^{n}

× \frac{(λq^{−2n}, bq^{−n}, λc/a, λdq^{−n}/a, λeq^{−n}/a, fq^{−n}, gq^{−n}, hq^{−n})_{n}}{(q, λq^{1−n}/b, aq^{1−2n}/c, aq^{1−n}/d, aq^{1−n}/e, λq^{1−n}/f, λq^{1−n}/g, λq^{1−n}/h)_{n}} ,

c_{n} = \frac{(aq^{1−2n}, bq^{n}/a, fq^{−n}, gq^{−n}, hq^{−n}, bq/f, bq/g, bq/h)_{∞}}{(b^{2}q/λ, λq^{−n}/b, aq^{1−2n}/c, aq^{1−n}/d, aq^{1−n}/e, aq^{1−n}/f, aq^{1−n}/g, aq^{1−n}/h)_{∞}}

× \frac{(λc/a, λdq^{−n}/a, λeq^{−n}/a, abq^{1−n}/λc, abq/λd, abq/λe)_{∞}}{(bcq^{n}/a, bd/a, be/a, bf/a, bg/a, bh/a)_{∞}} .

Using the simplifications (2.4) and (2.6), we get a_{n}/δ_{n} = α_{n}, b_{n}/δ_{n} = β_{n} and c_{n}/δ_{n} = γ_{n}, which yields (3.6). □

Let n → ∞ in (3.6), assuming |q^{2}a^{3}/bdefgh| = |c| < 1 and |λc/a| = |aq/de| < 1 while appealing to Tannery’s theorem [6] for being able to interchange limit and summation. One gets the following transformation formula

8ψ_{8}[\begin{matrix}q\sqrt{a}, −q\sqrt{a}, b, d, e, f, g, h \\ \sqrt{a}, −\sqrt{a}, aq/b, aq/d, aq/e, aq/f, aq/g, aq/h \end{matrix}; q, c]

= \frac{(aq, q/a, λc/a, aq/λd, aq/λe, b/a, bf/λ, bg/λ, bh/λ, λq/f, λq/g, λq/h)_{∞}}{(λq, q/λ, c, q/d, q/e, b/λ, bf/a, bg/a, bh/a, aq/f, aq/g, aq/h)_{∞}}

×_{8}ψ_{8}[\begin{matrix}q\sqrt{λ}, −q\sqrt{λ}, b, λd/a, λe/a, f, g, h \\ \sqrt{λ}, −\sqrt{λ}, λq/b, aq/d, aq/e, λq/f, λq/g, λq/h \end{matrix}; q, \frac{λc}{a}]

+ \frac{b}{a} \frac{(q, q/a, c/b, aq, bq/c, bq/d, bq/e, bq/f, bq/g, bq/h, d, e, f)_{∞}}{(q/b, c/a, b^{2}q/a, aq/b, aq/c, bd/a, be/a, bf/a, bg/a, bh/a, aq/d, aq/e, aq/f)_{∞}}

× \frac{(g, h)_{∞}}{(aq/g, aq/h)_{∞}} _{8}ϕ_{7}[\begin{matrix}b^{2}/a, bq/\sqrt{a}, −bq/\sqrt{a}, bd/a, be/a, bf/a, bg/a, bh/a \\ b/\sqrt{a}, −b/\sqrt{a}, bq/d, bq/e, bq/f, bq/g, bq/h \end{matrix}; q, c]

+ \frac{(q, q/a, b/a, aq/λd, aq/λe, λc/ab, aq, f, g, h)_{∞}}{(c, c/a, q/b, q/d, q/e, qb/λ, b^{2}q/λ, aq/f, aq/g, aq/h)_{∞}}

× \frac{(λc/a, λd/a, λe/a, bq/f, bq/g, bq/h, abq/λc, abq/λd, abq/λe)_{∞}}{(bd/a, be/a, bf/a, bg/a, bh/a, λ/b, aq/c, aq/d, aq/e)_{∞}}

×_{8}ϕ_{7}[\begin{matrix}b^{2}/λ, bq/\sqrt{λ}, −bq/\sqrt{λ}, bd/a, be/a, bf/λ, bg/λ, bh/λ \\ b/\sqrt{λ}, −b/\sqrt{λ}, abq/λd, abq/λe, bq/f, bq/g, bq/h \end{matrix}; q, \frac{λc}{a}] , (3.7)

where c = q^{2}a^{3}/bdefgh, λ = qa^{2}/cde, |c| < 1 and |λc/a| < 1. Note that when λ = a or when b = a, this identity is trivial. On the other hand, identity (3.7) can be derived from classical known identities as follows : first, apply [9, (5.6.1)] to both _{8}ψ_{8} series of (3.7), then the use of the transformation [9, (2.10.1)] gives some cancellations, and the remaining identity is finally a special case of the theta function identity [9, Ex. 5.22]. Aknowledgments. We thank Michael Schlosser for his valuable comments and pointing out a mistake in a previous version of this paper.

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Institut Girard Desargues, Université Claude Bernard (Lyon 1), 69622 Villeurbanne Cedex, France

E-mail address: jouhet@igd.univ-lyon1.fr
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