CURING The ANDREWS SYNDROME

Shalosh B. EKHAD^{1} and Doron ZEILBERGER^{2}

Abstract: George Andrews’s recent challenge to automated identity-proving and the WZ method is dealt with. It is argued that the rivalry between the classical and automated approaches to hypergeometric sums is beneficial to both.

AMS No. 05A15, 33S30, 39A00.

Kewords: Automated identity-proving, hypergeometric sums, semi-rigorous proofs.

PREFACE; In Praise of Bugs

Two summers ago, at the Ann Arbor third meeting on Algebraic Combinatorics (June 1994), and also at the Taormina conference in honor of Adriano Garsia (Aug. 1994), George Andrews described, rather dramatically, some apparent shortcomings of the WZ method (see [PWZ] for a description of this method). At one moment, he displayed the printout of the computer-generated proof of a certain hypergeometric identity, that was outputted by the Maple package EKHAD (that now accompanies [PWZ]). This necessitated 14 transparencies, taped together, which Andrews rolled down on the floor, producing an obvious comic effect.

The identity was ((1.6) of [A1]):

5F_{4} ({−2i − 1, x + 2i + 2, x − z + 1/2, x + i + 1, z + i + 1 \atop (x + 1)/2, x/2 + 1, 2z + 2i + 2, 2x − 2z + 1}; 1) = 0 ,  (MRR)

where we use the standard hypergeometric notation (see [GKP],[PWZ]).

In fact, the printout that was displayed was not even the proof of the identity itself, that was beyond the capabilities of EKHAD at the time, but the proof of a certain special case, obtained by specializing the parameter z.

As it turned out, EKHAD’s failure to give a complete proof of (MRR) at the time, as well as the length of the proof of the special case, were not the algorithm’s fault, but were due to a typo in the Maple coding of the second author^{3}. Once this typo was corrected, the proof of (MRR) that EKHAD outputs is rather short (it fits in less than one transparency), and is produced relatively

fast. Both the input and output files can be viewed in http://www.math.temple.edu/~zeilberg/synd.html,

the present paper’s Web Page.

This was a lucky mistake, since it lead Andrews to develop an elegant new technique, that he dubbed ‘Pfaff’s method’([A1-3]). Using this new technique, he not only proved (MRR), but in the process was naturally lead to the discovery and proof of nineteen additional brand new hypergeometric identities. These identities would never have been discovered had EKHAD been perfect two years ago.

Aug. 11, 1996. This paper is accompanied by a Web Page: http://www.math.temple.edu/~zeilberg/synd.html .

1 Supported in parts by the NSF.

2 Supported in part by the NSF.

3 It escaped notice before because it occurred in a rarely visited part of the program.

After the current version of EKHAD effortlessly proved identity (MRR), we were sure that it would just as easily prove all these other identities. This was implicit in the second author’s talk in Herb Wilf’s birthday conference, that took place in Philadelphia, June 12-15, 1996. When George Andrews asked us whether these new identities are equally provable by EKHAD, in real time, we replied that while we did not try yet, we were sure that they would pose no problem.

Imagine our surprise, and slight embarrassment, when the new, improved, and debugged, version of EKHAD ran out of memory in each and every one of Andrews’s twenty new identities, except for two (the original (4.2) (which is (MRR)) and (4.20) of [A1]). When one specializes the auxiliary parameters x and z to be specific rational numbers, then the output is ready after a few seconds. When only one of the parameters (x or z) is specialized, it takes about an hour of CPU time (on a Sun SPARC), but when both x and z are left alone, Maple ends with a memory fault (on our system).

In this paper we will outline how to get around this problem, and how to produce absolutely rigorous WZ-proofs for each of these 18 identities (and possibly many others yet to be discovered, on which EKHAD might fail for lack of memory), using commonly available computers. However, while it is feasible to get such a rigorous proof in each case, it is very time-consuming, and takes about a week running on our system on nice. On the other hand, the very same method also produces semirigorous (see [Z]) proofs very fast. Since we know that the identities are true (even if Andrews did not have a proof), and we know that we can find a complete proof, if we only were willing to wait a week/identity, we don’t see the point of actually wasting the Temple Math Department’s computer resources.

Any of our readers who wishes to have a complete proof is welcome to download our Maple package SYND, as well as any of the input files given in this paper’s Web Page, on their own computer. Alternatively, we would be happy to run the program for a fee of $300/identity. Once the fee is paid, the program would be run, and once it terminates, the output would be published in the Web Page, and the name of the donor would be prominently displayed. This is a good way to honor friends and relatives, by naming the proof after them. Instructions on how to order a proof are given in the Web Page mentioned above.

In order to demonstrate feasibility, we have run the program SYND on one of Andrews’s new identities: (4.4) of [A1]. The input and output files can be gotten from our Web page.

A Short Review of Creative Telescoping and WZ-certification

Let F (n, k) = F (n, k, c_{1}, …, c_{r}) be proper hypergeometric (see [PWZ], p. 64 for definition) in all its arguments. The Fundamental theorem of [PWZ] guarantees that there exist a non-negative integer J, and polynomials a_{j}(n) = a_{j}(n, c_{1}, …, c_{r}), j = 0, …, J free of k, as well as another proper hypergeometric term G(n, k) = G(n, k, c_{1}, …, c_{r}), such that G/F is a rational function in all its arguments, and such that:

∑_{j=0}^{J} a_{j}(n)F (n + j, k) = G(n, k + 1) − G(n, k) .  (CT)

It follows immediately, by summing w.r.t. k, that the definite sum:

A(n) = A(n, c_{1}, …, c_{r}) := ∑_{k} F (n, k, c_{1}, …, c_{r})

satisfies the linear recurrence equation with polynomial coefficients

∑_{j=0}^{J} a_{j}(n)A(n + j) = 0 .  (Recurrence)

Hence, in order to prove a conjectured identity

A(n) = B(n),

where B(n) is a certain explicitly given expression, all we have to do is verify that B(n) also satisfies the same recurrence, i.e.

∑_{j=0}^{J} a_{j}(n)B(n + j) = 0 ,

then check that the leading coefficient a_{J}(n) does not vanish at positive integers (if it does we just begin at the highest integer root, and check empirically the finite number of cases before). Finally we check that A(j) = B(j) for j = 0, …, J − 1. The identity A(n) = B(n) then would follow, by induction, for every positive integer value of n.

The recurrence and certificate, G(n, k), featured in (CT) can be found with the Gosper-Zeilberger algorithm ([PWZ][GKP]).

In this algorithm, in addition to the unknowns a_{0}, …, a_{J} one is also looking for the coefficients, b_{i}, of a certain polynomial:

b(k; n, c_{1}, …, c_{r}) = ∑_{i=0}^{K} b_{i}(n, c_{1}, …, c_{r})k^{i}

such that equation (6.1.3) of [PWZ], p. 108, holds, i.e.

p_{2}(k)b(k + 1) − p_{3}(k − 1)b(k) = p(k) ,  (G − Z)

for certain explicit polynomials p(k), p_{2}(k), p_{3}(k) that also depend on n, c_{1}, …, c_{r}, and that are derivable from the summand F (n, k). Furthermore p(k) involves the unknown coefficients of the recurrence a_{j}(n) linearly.

The next step is to expand (G − Z) in powers of k and compare the coefficients, getting a homogeneous system of linear equations in the unknowns a_{0}, a_{1}, …, a_{J}, b_{0}, …, b_{K}. If the system has no non-trivial solutions, it means that J was too low, so we try again with J ← J + 1. We are guaranteed to succeed eventually by the Fundamental Theorem (see [PWZ]).

Until Andrews’s paper [A1], the above was as easily said as done, and EKHAD had no difficulty in proving any identity of the above form. Sure, the largest contemporary computer is probably unable to find the recurrence that

A_{n} := ∑_{k=0}^{n} \binom{n}{k}^{100000}

satisfies, but for single sum summations with conjectured (or already known) explicit expressions, it never failed, as far as we know. Imagine our chagrin (and George Andrews’s justified glee) at the failure of our program on his new identities. We should remark that the failure was not theoretical but practical: insufficient memory to handle the large objects encountered.

The Bottle-Neck that Causes the Andrews Syndrome

As we saw above, the heart of the Gosper-Zeilberger algorithm is the solving a certain system of linear equations with symbolic coefficients. When the summand F (n, k) only depends on n and k, and the extra arguments (parameters) c_{1}, c_{2}, …, c_{r} are absent, then the entries of the coefficient matrix of the system are polynomials in the single variable n. However, when other parameters are present (in Andrews’s case we have x and z in addition to n), then the entries are rather large polynomials of several variables (in Andrews’s case of the three variables n, x, z).

It turns out that, say for (4.4) of [A1], the number of unknowns is 8: a_{0}, a_{1}, a_{2}, a_{3} (the coefficients of the recurrence operator) and b_{0}, b_{1}, b_{2}, b_{3} (the coefficients of the polynomial p(k)). In order to find the recurrence operator a_{0} + a_{1}N + a_{2}N^{2} + a_{3}N^{3} (where N is the forward shift in the variable n: Nf(n) := f(n + 1)), and the polynomial b(k) = b_{0} + b_{1}k + b_{2}k^{2} + b_{3}k^{3}, we have to solve a certain linear system:

M (a_{0}, a_{1}, a_{2}, a_{3}, b_{0}, b_{1}, b_{2}, b_{3})^{T} = 0 ,  (LinearSystem)

where M is an 8×8 matrix whose entries are certain (rather large) polynomials of the three variables n, x, z. As we said above, solving this system is beyond Maple’s capabilities running on our system. Our readers are welcome to try EKHAD on any of Andrews’s identities on their system. The input files for EKHAD (that failed on our system, but may work on bigger ones) are given in this paper’s Web page. We would be happy to hear of reports of any successful run, and we would announce them on that very same Web page.

A general WZ procedure for lowering the order of a recurrence by 1

Suppose ∑_{k} F (n, k) = r(n) is some identity that we wish to prove. Recall the WZ paradigm, which says to divide by the right hand side, and to try to prove instead the identity f(n) := ∑_{k}(F (n, k)/r(n)) = 1. Well, here is another possible advantage of this paradigm, which was pointed out by Herb Wilf: since the recurrence formula that the left side satisfies must have the solution f(n) = 1, it follows that the recurrence operator will always have a right factor of N − 1, where N is the forward shift in n. That being the case we can look for the recurrence that is satisfied by (N − 1)f(n), and it will be of one lower order than the original. To look for that recurrence, let’s rename the summand F (n, k)/r(n), and call it F (n, k) again. Then what we want to do is to apply the Zeilberger algorithm to the summand F (n+1, k)−F (n, k), instead of applying it to F (n, k) itself. We obtain a recurrence of order 1 less than we would otherwise have found, and we then need to show that it has only the trivial solution, which we do by displaying enough initial zero values. This procedure is perfectly general, and it applies in any situation where the form of the conjectured sum is known.

Getting Around the Andrews Syndrome

Consider again an identity of the form

∑_{k} F (n, k, c_{1}, …, c_{r}) = B(n, c_{1}, …, c_{r}) .

As above, we make the right side 1, by dividing by B(n). Renaming F ← F/B, we have to prove an identity of the form (suppressing the dependence on c_{1}, …, c_{r}):

A(n) := ∑_{k} F (n, k) = 1 .

It is easy to check whether A(0) = 1. hence it suffices to prove that

Ã(n) := A(n + 1) − A(n) := ∑_{k}[F (n + 1, k) − F (n, k)] = 0 .  (Efes)

In 99% of the cases, one can apply Gosper’s algorithm to the summand F (n + 1, k) − F (n, k) (with respect to k, see [PWZ], chapter 7), and get another hypergeometric term G(n, k) (which furthermore can be written as R(n, k)F (n, k), where R(n, k) is a rational function), such that

F (n + 1, k) − F (n, k) = G(n, k + 1) − G(n, k) .  (W Z)

This is the WZ-equation. Summing (W Z) with respect to k immediately yields (Efes). Summing (W Z) with respect to n yields the so-called companion identity (see ibid.).

However, in the remaining %1 of the cases, that include Andrews’s twenty identities discussed here, Gosper’s algorithm tells us that no such (hypergeometric) G(n, k) exists. The more general Gosper-Zeilberger algorithm guarantees, however (writing F̃(n, k) := F (n + 1, k) − F (n, k)) that there exist a non-negative integer J, and a_{0}, …, a_{J}, and G̃(n, k) such that

∑_{j=0}^{J} a_{j}(n)F̃(n + j, k) = G̃(n, k + 1) − G̃(n, k) .  (CT′)

As before, summing with respect to k, we get:

∑_{j=0}^{J} a_{j}(n)Ã(n + j) = 0 .  (Recurrence′)

Once these are found it would follows that Ã(n) ≡ 0 provided it is so for n = 0, …, J − 1.

Note that the ‘J’ of (Recurrence′) is one less than that of (Recurrence), since considering Ã(n) rather than A(n) reduces the order of the recurrence by 1.

While the above process reduces the number of unknowns by 1, the resulting symbolic linear system still proves too big for Maple to handle. And now comes the second twist on the Gosper-Zeilberger algorithm. In order to prove that Ã(n) ≡ 0, we don’t actually have to know what the a_{0}, a_{1}, …, a_{J} and the b_{0}, …, b_{K} are exactly! All we have to know is that they exist!, plus we have to make sure that the leading coefficient a_{J}(n) of the recurrence does not vanish on any positive integer value of n (or if it does, then we have to know the highest such value).

The existence of a non-trivial solution of the homogeneous linear system (LinearSystem) (when it now applies to the modified sum as above), is exactly equivalent to the fact that the determinant of the square matrix M vanishes. This determinant is a polynomial in the variables n, c_{1}, …, c_{r}. If we can find a priori bounds for the degrees in each of its variables, then plug in enough special cases (the number of which should be at least equal to the product of 1 plus the degrees in each of the variables) into this determinant, and evaluate these numerous, but fast-to-compute resulting numerical determinants, and check whether they are always 0, we would have a completely rigorous proof.

As hinted in [Z], if you take a non-zero polynomial out of the blue and plug in random values, it is extremely unlikely that it would be zero. It such a polynomial yields zero for, say, a hundred different tries, then it is much more likely that the entire framework of mathematics is flawed than that this particular polynomial is non-zero. Hence we have a situation where in order to prove an identity we have the costly, but feasible, option of a completely rigorous proof, and an extremely fast and inexpensive way to prove it with probability 1 − ϵ. In the package SYND, that comes with this paper, and that can be downloaded from its Web Page, the procedure PROOF2 has Certainty as one of its arguments, that can be adjusted by the user. Setting Certainty:= 1 yields a rigorous, but time-consuming, proof. Setting it, to say 0.1 would already yield a 99.9999%-sure proof very fast.

So now we know (either for sure, or almost surely) that the a_{0}(n, c_{1}, …, c_{r}), …, a_{J}(n, c_{1}, …, c_{r}), and the G(n, k) in Eq. (CT′) exist, and hence Eq. (Recurrence′) holds, for some a_{0}, …, a_{J}. We don’t really care what they are except, as mentioned above, we have to make sure that a_{J}(n, c_{1}, …, c_{r}) does not vanish on positive integers (and if it does, then to find the largest such). But this is easy and fast. If a_{J}(n, c_{1}, …, c_{r}) would have vanished at n = n_{0}, then (n − n_{0}) would have been a factor, and it still would have been a factor of a_{J}(n, c_{1}^{0}, …, c_{r}^{0}), for any specialization c_{1} = c_{1}^{0}, …, c_{r} = c_{r}^{0}. So all we have to do is apply procedure ct or zeil of EKHAD to any such specialization, and make sure that the polynomial a_{J} does not have a factor of the form (n − n_{0}). Now the summand only depends on two symbols: n, and k, and ct runs very fast (at least on Andrews’s identities discussed here).

A Priori Bounds For the Degrees of the Determinant

In order to prove that a polynomial P (x_{1}, …, x_{m}) is the zero polynomial, by plugging in sufficiently many values for its arguments, we need a priori bounds for its degrees in each and every one of its arguments. If its degree in x_{i} is d_{i}, then checking at all the integer points −(d_{i} + 1)/2 ≤ x_{i} ≤ (d_{i} + 1)/2 (i = 1, …, m), and verifying that the value of P is zero at each of these points, would constitute a rigorous proof that P is the zero polynomial.

In the present scenario, the polynomial P (x_{1}, …, x_{m}) is given as a determinant P := det(a_{i,j}(x_{1}, …, x_{m})), where (a_{i,j}) is a square matrix. To get an upper bound d_{i}(usually very dull), for the degree of the determinant in the variable x_{i}, we replace each entry a_{i,j} by its leading term with respect to x_{i}. For example 4x^{3}z^{5} + 3x^{2}z^{7} + 5xz would be replaced by 4x^{3}x^{5} when the degree in x is sought, and by 3x^{2}z^{7} when the degree in z is desired. Next, we take the permanent. Since the permanent of a matrix with monomial entries can’t have any cancellations, the degree in the variable x_{i} of the resulting permanent would definitely constitute an upper bound for d_{i}.

The Package Synd

The present method (or rather the present twist on the old method) is implemented by the Maple package SYND, that accompanies this paper. It can be downloaded from either

http://www.math.temple.edu/~zeilberg/synd.html (download SYND), or from
ftp://ftp.math.temple.edu/pub/zeilberg/programs, or by anonymous ftp to ftp.math.temple.edu

(login as anonymous, password as instructed (your E-mail address), then cd pub/zeilberg/programs <CR>, followed by, get SYND <CR>.To exit ftp you type: quit <CR>.

Once you have SYND on your own computer, make sure that you have Maple (if you don’t, get it!), and that you are in the directory where SYND is. Then get into Maple by typing maple <CR>. Once inside Maple, type read SYND; <CR>, and get the on-line help by typing ezra();CR.

The two main procedures are PROOF1 and PROOF2. The former handles sums with one extra parameter, while the latter handles sums, like those of Andrews[A1], with two extra parameters (x and z). We did not bother to write the analogous procedures for more auxiliary parameters, but the Maple-literate reader can do it with no trouble.

The function call for PROOF1 is:

PROOF1(SUMMAND,RHS,k,n,LowerLimit,UpperLimit,a,Certainty);

Here SUMMAND is the hypergeometric summand, given either in terms of factorials, or binomial coefficients, or rf, where rf(a,k) is the raising factorial (a)_{k} := a(a + 1) ⋯ (a + k − 1); RHS is the conjectured right hand side; k is the (single-) summation variable; n is the variable with respect to which the recurrence is desired; LowerLimit and UpperLimit are respectively where the summation starts and ends; a is the auxiliary parameter; Finally, Certainty is the rigor-level. Setting Certainty:=1;, would yield a rigorous proof. Setting it to anything less would give a semi-rigorous proof. Be warned that Certainty=0.1 does not mean that the probability that the identity is true is %10! It is probably true with probability %99.999999. The meaning of Certainty is the fraction of trials that we attempt, compared to the full intervals in n and in a, that is needed for a rigorous proof.

Example: While EKHAD can handle the Chu-Vandermonde identity

∑_{k=0}^{n} \binom{n}{k}\binom{a}{k} = \binom{n + a}{n} ,

in a few seconds, just for the sake of example, here is the function call that does it in SYND. Since Certainty is set to 1, this would yield a completely rigorous proof.

PROOF1(binomial(a,k)*binomial(n,k),binomial(a+n,a),k,n,0,n,a,1); .

PROOF2 is exactly as above, except that we have to specify the two auxiliary parameters: x, and z.

The function call for PROOF2 is:

PROOF2(SUMMAND,RHS,k,n,LowerLimit,UpperLimit,x,z,Certainty); .

where x and z are the auxiliary parameters. For example the Dixon identity

∑_{k=−n}^{n}(−1)^{k}\binom{a + b}{a + k}\binom{a + n}{n + k}\binom{b + n}{b + k} = \frac{(a + b + n)!}{a!b!n!} ,

is proved, by SYND, with the following call:

PROOF2((-1)**k*binomial(a+b,a+k)*binomial(a+n,n+k)*binomial(b+n,b+k), (a+b+n)!/a!/b!/n!,k,n,
-n,n,a,b,1);

(Once again this is only for the sake of example, since EKHAD does it effortlessly).

Andrews’s 20 Identities

The input files for all the identities of section 4 of [A1], as well as the corresponding output files (with Certainty=0.1), can be retrieved from the Web page. As we said above, the only identity for which we ran the program with Certainty=1 is identity (4.4). Readers who have Maple and have idle time to spare, are welcome to download any or all the input files, change the last argument of PROOF2 to 1 (instead of 0.1), and run it on their computer. We request that you notify us, so that we can announce that the WZ-style rigorous proof of the given identity has been performed, with due acknowledgement to the computer and the human owner.

Lily Yen’s Method

Another possibility, that might work, of getting around the Andrews syndrome is to adapt Lily Yen’s[Y] beautiful approach. Yen found an a priori bound, L , easily derivable from the identity, such that if the identity is true for n = 0, 1, …, L, then it would be true for all n. Unfortunately, it is so enormous that at present it only has theoretical interest. However, it seems to us that an appropriate modification to the situation described in the present paper, where one has extra parameters, would yield quite small and practical upper bounds.

The reason for the gargantuan size of Yen’s upper bound is having to bound the largest integer root of the leading coefficient a_{J}(n). The upper bound for the order of the recurrence, J, is quite small. Since now we have extra parameters, we can rule out the possibility of positive integer roots as above, by running ct of EKHAD on a specialization. It would be interesting and useful to make this precise, and to implement it.

Peter Paule’s Method

Andrews’s identities are also interesting from the WZ theory point of view. The orders of the recurrences outputted by EKHAD (3 and 2) are two higher than the orders of the minimal recurrences (1 and 0) satisfied by the right sides.

They provide many new non-trivial examples to the phenomenon described on p. 117 of [PWZ]. This phenomenon, of not getting the minimal recurrence, is much more widespread for q-series, in which Peter Paule[P] introduced a very useful order-reducing preprocessing device. Paule’s method is also useful for ordinary hypergeometric sums.

At present it is not clear how to apply Paule’s method to Andrews’s sums, but we suspect that an appropriate generalization will do the job. Perhaps Andrews’s identities are limiting cases of more general identities that come from multiple sums, on which there would be some obvious symmetry group with respect to which one would be able to apply Paule-symmetrization.

Conclusion

Our Wise men, let their memory be blessed said: ‘Kinat Sofrim Tarbe Khochma’, which, roughly means: ‘Rivalry among scholars increases knowledge’. In the present case the rivalry is between human and machine. While the meta-mathematical debate engendered in [Z] and [A4] is unlikely to be resolved in our time, mathematics proper does benefit.

The second moral is that mistakes are crucial for progress. According to the current dogma in biology, we humans would have still been amoebas if not for a series of lucky mistakes in biological transmission of information. The same can be said for science, and even for mathematics. If the previous version of EKHAD did not contain a bug, George Andrews would have obtained the proof of (MRR) right away, and would never have needed to find another proof, that lead to twenty new beautiful identities. While the WZ method is also capable of discovering (and proving at the same time) new identities out of old ones, we do not know at present how to rediscover, naturally, Andrews’s identities from scratch.

Bruno Salvy has told us that the question of deciding how many random checks of the vanishing of a polynomial are needed in order to deduce its nullity, within a prescribed margin of error, has been dealt with. See his message[B] that can be accessed from this paper’s Web page.

One last point. The present method is easy to parallelize. This is because it boils down to checking many special instances, that can be done independently of each other. If the need would arise in the future to prove a huge identity (for example because it would imply the Riemann Hypothesis), then it would be feasible to have an International collaboration of many computers.

Acknowledgement

We wish to thank George Andrews and Herb Wilf for many stimulating discussions.

REFERENCES

[A1] George Andrews, Pfaff’s Method (I): The Mills-Robbins-Rumsey determinant, Discrete Mathematics (Garsia Festschrifft issue), to appear.

[A2] George Andrews, Pfaff’s Method (II): diverse applications, to appear.

[A3] George Andrews, Pfaff’s Method (III): comparison with the W-Z method, Elect. J. of Combinatorics 3(2)(1996) [Foata Festschrifft], R21.

[A4] George Andrews, The death of proof? Semi-rigorous mathematics? you’ve got to be kidding!, Math. Intell. 16, no. 4, 16-18 (Fall 1994).

[B] Bruno Salvy, Message to Doron Zeilberger, published, by permission, in this paper’s Web page

http://www.math.temple.edu/~zeilberg/synd.html.

[GKP] Ronald Graham, Donald Knuth, and Oren Patashnik, “Concrete Mathematics”, second edition, Addison-Wesley, Reading, 1994.

[P] Peter Paule, Short and easy computer proofs of the Rogers-Ramanujan identities and of identities of similar type, Elect. J. of Combinatorics 1(1994) R10.

[PWZ] Marko Petkovsek, Herbert Wilf, and Doron Zeilberger, “A=B”, AK Peters, Wellesley,

(1996). Web Page:  http://central.cis.upenn.edu/~wilf/AeqB.html .

[Y] Lily Yen, A two-line algorithm for proving terminating hypergeometric identities, J. Math. Anal. Appl. 198(1996), 856-878.

[Z] Doron Zeilberger, Theorems for a price: Tomorrow’s semi-rigorous mathematical culture, Notices of the Amer. Math. Soc. 40 # 8, 978-981 (Oct. 1993). Reprinted: Math. Intell. 16, no. 4, 11-14 (Fall 1994). ——————- Shalosh B. Ekhad, Department of Mathematics, Temple University, Philadelphia, PA 19122.

E-mail:  ekhad@math.temple.edu ; WWW: http://www.math.temple.edu/~ekhad .

Doron Zeilberger, Department of Mathematics, Temple University, Philadelphia, PA 19122.

E-mail:  zeilberg@math.temple.edu ; WWW: http://www.math.temple.edu/~zeilberg .