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\centerline
{\bf  Bijective and Automated Approaches to Abel Sums }
\bigskip
\centerline
{\it Gil KALAI and Doron ZEILBERGER}


\qquad \qquad {\it  Dedicated to Dominique Foata (b. Oct. 12, 1934), on his forthcoming 90th birthday}

{\bf Abstract}: In this tribute to our guru, Dominique Foata, we return to one of our (and Foata's) first loves, and revisit 
Abel sums and their identities, from two different viewpoints.


{\bf Preface}

In the very first issue of {\it Crelle's journal} (the first mathematical periodical solely dedicated to mathematics), Niels Henrik Abel published
a two-page paper [A], stating and proving his eponymous identity. This led to an intensive study of general {\it Abel Sums} by many people, 
(see [R] [C] and their numerous references), and
to beautiful bijective approaches pioneered by Dominique Foata and Aim\'e Fuchs [Fo][FoFu] that led to Fran{\c c}on's elegant proof [Fr] (see [C], p. 129).
This tribute consists of two independent parts. The first part is {\it bijective}, while the second
part is {\it automated}, elaborating and extending  John Majewicz's 1997 PhD thesis [M1][M2], and more importantly,
fully implementing it (and its extension) in a Maple package   \hfill\break
{\tt https://sites.math.rutgers.edu/\~{}zeilberg/tokhniot/AbelCeline.txt} \quad .

(Majewicz's original Maple code, unfortunately, was lost).


{\bf Part I: Bijective proofs (\`a la Foata) of an Abel-type identity and a generalization}

In [Ka] (see also [Ka']), the first author proved (as a special case of  more general results) the following Abel-type identity. Let $n,p$ by nonnegative integers, then

$$
\sum _{k=0}^n {{n} \choose {k}} k^k (n-k)^{n-k+p} = \sum _{k=0}^n {{n} \choose {k}}n^k (n-k)! S(p+n-k,n-k) \quad .
\eqno(1)
$$

Here, $S(n,k)$ are the Stirling numbers of the second kind, and $k! S(n,k)$ is the number of maps from a set of size $k$ onto a set of size $n$. (This property can serve as the definition of these numbers.) 

A special case of this formula is attributed to Cauchy (Equation (24) in Chapter 1 of [R]):

$$
\sum _{k=0}^n {{n} \choose {k}} k^k (n-k)^{n-k} = \sum _{k=0}^n {{n} \choose {k}}n^k (n-k)! \quad .
\eqno(2)
$$


Here we present a combinatorial proof, in the style of Foata [Fo][FoFu] for Formulas $(2)$ and $(1)$. 
We use the notation $[n]=\{1,2,\dots,n\}$, and to make the argument clearer we will present both proofs.

\vfill\eject

{\bf The Proofs}

{\bf Proof of Equation $(2)$}:
The left hand side counts triples $f,A,B$ where
$f$ is a function from $[n]$ to $[n]$
and the following conditions hold:

$$
A \cup B=[n], A \cap B =\emptyset, f(A)\subset A {\rm ~and~}  f(B) \subset B.
\eqno(3)
$$


Indeed, if $|A|=k$, there are ${{n} \choose {k}}$
ways to choose $A$ ($B$ is now determined, $B=[n] \backslash A$), $k^k$ ways to choose the restriction of $f$ to $A$ subject to the condition $f(A) \subset A$ and $(n-k)^{n-k}$ ways to choose the restriction of $f$ to $B$ subject to the condition $f(B) \subset B$.  

The right hand side counts triples $f,C,D$ where
$f$ is a function from $[n]$ to $[n]$
and following conditions hold:

$$
C \cup D=[n], C \cap D =\emptyset, f(D) = D.
\eqno(4)
$$
(Here, 
$f(D)=D$ means that $f$ is a bijection from $D$ to $D$; Note that we relaxed the condition for $C$ compared to $A$ and strengthened the condition for $D$ compared to $B$.) Indeed, if $|C|=k$, there are ${{n} \choose {k}}$
ways to choose $C$, $n^k$ ways to choose the restriction of $f$ to $C$ (no conditions here) and  $(n-k)!$ ways to choose the restriction of $f$ to $D$ subject to the condition $f(D) =D$.

The crucial observation is:

$\bullet$ For {\it every} function $f$ from $[n]$ to $[n]$ the number of pairs $(A,B)$ that satisfy Equation $(3)$ equals the number of pairs $(C,D)$ that satisfy equation $(4)$. 

Indeed if $(A,B)$ is a pair that satisfies Equation $(3)$, we can take 
$$
D=f(f(\cdots (f(B))\cdots ) {\rm ~and~} C=[n] \backslash D.
\eqno(5)
$$


In other words $D=d \in [n]$ such that for every $k \ge 0$ there exists $b \in B$ with
$d=f^k(b)$.
Taking the inverse operation $B=f^{-1}(f^{-1}(\cdots (f^{-1}(D))\cdots )$
(and $A=[n] \backslash B$) brings you from a pair $(C,D)$ to a pair $(A,B)$.


{\bf Proof of Equation $(1)$}:

Let $X=[n]=\{1,2,\dots. n\}$, and $Y=[n+p]$ 
Consider all triples $(f, A, B)$ where, this time, $f$ maps $[n+p]$ to $[n]$, and $A,B$ satisfy

$$
A \cup B=[n], A \cap B =\emptyset, f(A)\subset A {\rm ~and~}  f(B \cup [n+1,n+p]) \subset B.
$$

We also consider triples $(f, C, D)$ where, $f$ maps $[n+p]$ to $[n]$, and $C,D$ satisfy

$$ 
C \cup D=[n], C \cap D =\emptyset, f(D \cup [n+1,n+p]) = D.
\eqno(7)
$$

Also in this case a stronger statement holds: For every function  $f: [n+p] \to [n]$  there is a bijection between between pairs $(A,B)$ satisfying equation $(6)$ and pairs $(C,D)$ satisfying equation 
$(7)$ and the bijection is given again by $(5)$.


This bijection is implemented in the Maple package {\tt AbelBijection.txt}, available from

{\tt https://sites.math.rutgers.edu/\~{}zeilberg/tokhniot/AbelBijection.txt} \quad .

See the front of this article \hfill\break
{\tt https://sites.math.rutgers.edu/\~{}zeilberg/mamarim/mamarimhtml/abelKZ.html} for a sample input file and its corresponding output file.

{\bf Remark:} If we give each element $k \in [n]$ a weight $w_k$ and a function $f$ from $[n+p] \to [n]$ the weight $w(f):=\prod \{w_{f(k)} \, : \, k \in [n]\}$ then our bijective proof 
gives a Hurwitz-type generalization of Abel's formula, see Hurwitz [H] and Excercise 30 in Section 2.3.4.4. of Knuth [Kn]. 
(For Formula $(1)$ we obtain a Hurwitz-type generalization of the Stirling numbers of the second kind.) 
 
{\bf Part II: Automating Abel Sums}

Abel's original identity had many proofs, but the one by Shalosh B. Ekhad and John Majewicz [EM] (written $30$ years ago, and dedicated to Dominique Foata on his $60^{th}$ birthday)
has the distinction that it was {\it computer-generated}, yet with a bit of patience,
{\it humanly readable} and verifiable. In order to motivate the sequel, let's reproduce it in its entirety.



 
{\bf Abel's identity}: For any non-negative integer $n$,
$$
\sum_{k=0}^n {{n} \choose {k}} \,(r+k)^{k-1}(s-k)^{n-k}=\frac{(r+s)^n}{r}  \quad .
\eqno(8)
$$
 
{\bf Proof}  Let $F_{n,k}(r,s)$ and $a_n(r,s)$ denote, respectively, 
the summand and sum on the LHS of (8), and let 
$G_{n,k}:=(s-n) {{n-1} \choose {k-1}} \,(k+r)^{k-1}(s-k)^{n-k-1}$. Since
$$
F_{n,k}(r,s)-sF_{n-1,k}(r,s)-(n+r)F_{n-1,k}(r+1,s-1)
        +(n-1)(r+s)F_{n-2,k}(r+1,s-1)=G_{n,k}-G_{n,k+1} \,,
$$
(check!), we have by summing from $k=0$ to $k=n$, thanks to the telescoping
on the right:
$$
 a_n(r,s)-sa_{n-1}(r,s)-(n+r) a_{n-1}(r+1,s-1)+
        (n-1)(r+s) a_{n-2}(r+1,s-1)=0.
\eqno(9)
$$
Since $(r+s)^n\cdot r^{-1}$ also satisfies 
this recurrence (check!) with the same initial 
conditions $a_0(r,s)=r^{-1}$ and 
$a_1(r,s)=(r+s)\cdot r^{-1}$, $(8)$ follows.\halmos

This proof was derived using John Majewicz's brilliant adaptation of {\it Sister Celine's technique} [Fa1][Fa2][Z] (see also [PWZ], chapter 4). Recall that Sister Celine was interested in finding {\it pure recurrence relations} of the form
$$
c_0(n)a_n+  c_1(n)a_{n+1}+  \dots + c_L(n) a_{n+L}=0 \quad,
\eqno(10)
$$
where $c_0(n), \dots, c_L(n)$ are polynomials in $n$ for sequences $a_n$, that are  defined by expressions of the form
$$
a_n:=\sum_{k=-\infty}^{\infty} F_{n,k} \quad,
$$
where  $F_{n,k}$ is proper hypergeometric (see [PWZ] for the definition, in particular,  $F_{n+1,k}/F_{n,k}$ and $F_{n,k+1}/F_{n,k}$ are both rational functions of $n$ and $k$) .
The way she did it was to search for a recurrence of the form: 
$$
\sum_{i=0}^{L} \sum_{j=0}^{M} b_{ij}(n) F_{n+i,k+j} \, = \, 0 \quad ,
\eqno(11)
$$
for  {\it some} positive integers $L$ and $M$.  (In her case by hand, but nowadays it has all been fully automated.)

Dividing by $F_{n,k}$, and clearing denominators, looking at the numerator,  and then setting all the coefficients of powers of $k$ to $0$, yields a system of {\it linear equations} (with coefficients that
are polynomials in $n$) for the {\it undetermined} $b_{ij}(n)$.

Having found the $b_{ij}(n)$, summing Eq. $(11)$ from $k=-\infty$ to $k=\infty$, yields Eq. $(10)$ with
$$
c_i(n)=\sum_{j=0}^{M} b_{ij}(n) \quad, \quad 0 \leq i \leq L \quad .
$$


In his  PhD thesis  [M1][M2] (written under the direction of the second author), John Majewicz adapted Sister Celine's method to {\it Abel Type} sums, of the form
$$
a_n(r,s)=\sum_{k=0}^n \,F_{n,k} \, (r+k)^{k-1+p}(s-k)^{n-k+q} x^k  \quad ,
\eqno(12)
$$
where $F_{n,k}$ is hypergeometric in $n$ and $k$. Here $p$ and $q$ are arbitrary integers, and $x$ is any number (or symbol). It is no longer the case that
$a_n(r,s)$ satisfies a {\it pure} recurrence in $n$, with $r$ and $s$ {\bf fixed}, but it does satisfy a {\it functional recurrence}, similar (but often much more complicated) to Eq. $(10)$.
Denoting the summand of $(12)$ by $\overline{F}_{n,k}(r,s)$
$$
\overline{F}_{n,k}(r,s) \, := \, F_{n,k} (r+k)^{k-1+p}(s-k)^{n-k+q} x^k  \quad ,
$$
one looks for polynomials $b_{ij}(n)$ (that also depend on $r,s,p,q$ and $x$, but are {\bf free} of $k$), such that
$$
\sum_{i=0}^{L} \sum_{j=0}^{M} b_{ij}(n) \overline{F}_{n+i,k+j}(r-j,s+j) \, = \, 0 \quad .
\eqno(13)
$$
Dividing by  $\overline{F}_{n,k}(r,s)$ (since $\overline{F}_{n+i,k+j}(r-j,s+j)/\overline{F}_{n,k}(r,s)$ is still a rational function of $n$ and $k$), clearing denominators, looking at the
numerator, and setting all the coefficients of powers of $k$ to $0$, we get again a system of linear equations for the undetermined quantities $b_{ij}(n)$. We then ask Maple
to kindly {\tt solve} them, and if {\it in luck} we get a non-zero solution. It can be shown ([M1][M2]) that one can always find orders $L$ and $M$ for which such a system
is solvable (for sufficiently large $L$ and $M$, there are more unknowns than equations). Having found such a recurrence for $\overline{F}_{n,k}$, summing over $k$ yields
a functional recurrence for $a_n(r,s)$.
$$
\sum_{i=0}^{L} \sum_{j=0}^{M} b_{ij}(n) a_{n+i}(r-j,s+j) \, = \, 0 \quad .
\eqno(14)
$$

{\bf Implementation}

We fully implemented the Celine-Majewicz algorithm in a Maple packge {\tt AbelCeline.txt}  available from

{\tt https://sites.math.rutgers.edu/\~{}zeilberg/tokhniot/AbelCeline.txt} \quad . The function call is

{\tt FindOpe(F,n,k,r,s,R,S,N,K,MaxOrd); } \quad,

where {\tt MaxOrd} is the maximal order of the recurrence you are willing to tolerate.

It inputs a hypergeometric term $F_{n,k}$ and outputs the recurrence in the form of the corresponding operator, where $N$, $R$, and $S$ are the forward shift operators in
$n$, $r$, and $s$, respectively. To get a computer-generated paper, in {\it humanese}, the function call is:

{\tt Paper(F,n,k,r,s,R,S,N,K,MaxOrd): } \quad .

{\bf Sample Output}

Typing

{\tt Paper(binomial(n,k)*x**k,n,k,r,s,2):},

yields in $0.12$ seconds the following deep fact.

Let, for any integers $p$ and $q$ and number (or symbol) $x$
$$
a_n(r,s) := \sum_{k=0}^{n} {{n} \choose {k}} (r+k)^{k-1+p}(s-k)^{n-k+q}\,x^k \quad,
$$
then
$$
a_{n}\! \left(r , s\right) = 
\left(n x +r x \right) a_{n -1}\! \left(r +1, s -1\right)+s a_{n -1}\! \left(r , s\right)+\left(-n r x -n s x +r x +s x \right) a_{n -2}\! \left(r +1, s -1\right) \quad .
$$

Typing

{\tt Paper(1/(k!**2*(n-k)!)*x**k,n,k,r,s,3):}

yields that the sequence of polynomials in $(r,s)$ defined by
$$
a_n(r,s) := \sum_{k=0}^{n} \frac{1}{k!^2(n-k)!} \, (r+k)^{k-1+p}(s-k)^{n-k+q}\,x^k \quad,
$$
satisfies the functional recurrence
$$
a_{n}\! \left(r , s\right) = 
\frac{x \left(n +r \right) a_{n -1}\! \left(r +1, s -1\right)}{n^{2}}+\frac{\left(2 n^{2}-2 n s -2 n +s \right) s a_{n -1}\! \left(r , s\right)}{\left(n -1-s \right) n^{2}}
$$
$$
-\frac{\left(n^{2}+2 n r -2 r s -s^{2}-n -r \right) x a_{n -2}\! \left(r +1, s -1\right)}{\left(n -1-s \right) n^{2}}
$$
$$
-\frac{\left(n -s \right) s^{2} a_{n -2}\! \left(r , s\right)}{\left(n -1-s \right) n^{2}}+\frac{\left(n r +n s -r s -s^{2}\right) x a_{n -3}\! \left(r +1, s -1\right)}{\left(n -1-s \right) n^{2}} \quad .
$$


To see numerous other examples, read  the output files in the front of this article:

{\tt https://sites.math.rutgers.edu/\~{}zeilberg/mamarim/mamarimhtml/abelKZ.html} \quad .


In particular, to see the functional recurrence satisfied by the `innocent' sum:
$$
a_n(r,s) := \sum_{k=0}^{n} {{n} \choose {k}}  {{n+k} \choose {k}} (r+k)^{k-1+p}(s-k)^{n-k+q}\,x^k \quad,
$$
see the output file

{\tt https://sites.math.rutgers.edu/\~{}zeilberg/tokhniot/oAbelCeline4.txt} \quad .

Notice how complicated things get!


{\bf Getting differential recurrences}

Since $r$ and $s$, as opposed to $n$ and $k$, are `continuous variables', and since the functional recurrences gotten by Majewicz' Abel-Celine technique get so
complicated even with very simple summands, it occurred to us to look for {\it differential} recurrences.  For any bi-variate (proper) hypergeometric term $F_{n,k}$, defining (as above)
$$
a_n(r,s) := \sum_{k=0}^{n} F_{n,k} (r+k)^{k-1+p}(s-k)^{n-k+q}\,x^k \quad,
$$
where $p$ and $q$ are integers (but may be left symbolic) and $x$ is a number (again, it can be left symbolic),
one looks for differential-recurrence equations of the form
$$
\sum_{i=0}^{L}\sum_{j=0}^{M} b_{i,j}(n,r,s) \frac{d^{i}}{dr^i} a_{n+j}(r,s) \, = \, 0 \quad ,
$$
and
$$
\sum_{i=0}^{L}\sum_{j=0}^{M} c_{i,j}(n,r,s) \frac{d^{i}}{ds^i} a_{n+j}(r,s) \, = \, 0 \quad ,
$$
Together these pairs of differential recurrence equations, combined with initial conditions, uniquely determine $a_n(r,s)$.

In fact, one can consider far more general `kernels'. We can find a pair of differential recurrence equations for sums of the form
$$
a_n(r,s) := \sum_{k=0}^{n} F_{n,k} K(r,s,n,k) \quad,
$$
for any {\it kernel} $K(r,s,n,k)$ such that $(\frac{\partial K}{\partial r})/K$, and  $(\frac{\partial K}{\partial s})/K$ are rational functions of $(r,s)$ (and $k$).

We proceed analogously. One applies the generic operator to the summand $\overline{F}(n,k)$, expands, clears denominators, looks at the numerator, then equates
all the powers of $k$ to $0$, getting a system of linear equations that Maple can {\tt solve} for you.

To get these pairs of equations, in verbose, human-readable from, type:

{\tt PaperD(F,n,k,r,s,MaxOrd,KER):}

where {\tt MaxOrd} is the  maximum order you are willing to tolerate, and {\tt KER} is the kernel. For example, typing

{\tt PaperD(binomial(n,k),n,k,r,s,2,(r+k)**(k-1+p)*(s-k)**(n-k+q)*x**k):} \quad,


yields in a fraction of a second the facts that the sequence of polynomials in $r$ and $s$, $a_n(r,s)$ (for any $p$ and $q$ and $x$)
$$
a_n(r,s) := \sum_{k=0}^{n} {{n} \choose {k}} (r+k)^{k-1+p}(s-k)^{n-k+q}\,x^k \quad,
$$
satisfy the pair of differential-recurrence equations (the first in $r$, the second in  $s$)
$$
-\left(p n +n s -n +p +s -1\right) a_{n}\! \left(r , s\right)+\left(n r +n s +r +s \right) \left(\frac{\partial}{\partial r}a_{n}\! \left(r , s\right)\right)
$$
$$
+(n+p)a_{n +1}\! \left(r , s\right)-\left(n +r +1\right) \left(\frac{\partial}{\partial r}a_{n +1}\! \left(r , s\right)\right)=0 \quad,
$$
and
$$
-\left(n +1\right) \left(q +n -s +1\right) a_{n}\! \left(r , s\right)+q a_{n +1}\! \left(r , s\right)+\left(n -s +1\right) \left(\frac{\partial}{\partial s}a_{n +1}\! \left(r , s\right)\right) \, = 0 \quad .
$$

Note that setting $x=1$ and $p=0,q=0$ yields yet-another (automatic) proof of the original identity $(8)$. In fact it is closer in spirit to Niels Abel's original proof, that also used
differentiation (or rather integration).

{\bf Remark:} The proof technique of this part can prove (2) and can be adapted (with some pre-processing) to prove (1).

To see many other examples, look at the output files

{\tt https://sites.math.rutgers.edu/\~{}zeilberg/tokhniot/oAbelCeline5.txt} \quad ,

and

{\tt https://sites.math.rutgers.edu/\~{}zeilberg/tokhniot/oAbelCeline6.txt} \quad .

To see examples with more complicated kernels, see

{\tt https://sites.math.rutgers.edu/\~{}zeilberg/tokhniot/oAbelCeline7.txt} \quad .

Readers can easily find many other such deep  facts by playing with {\tt AbelCeline.txt}. Enjoy!


{\bf Comment}: The anonymous referees and the non-anonymous editor (Christian Krattenthaler) made the following interesting remarks. We thank them
for allowing us to quote them {\it verbatim}.

{\it The paper has a nice and interesting observation
(namely that in some cases it might be more fruitful to consider
differential operators instead of difference operators).
However, the corresponding theory has already been developed in very
general form by Frederik Chyzak, Manuel Kauers, and Bruno Salvy in the interesting paper [CKS],

In this general form (Ore algebras) the corresponding algorithms
have been implemented in Christoph Koutschan's Mathematica package. 
It may however be that the present paper's  specialized implementation is faster and more
efficient than the general-purpose implementation of Koutschan.

It also seems that one may be able to get differential recurrences, where one takes {\bf both} differentiations with respect to $r$ and with respect to $s$.
}


{\bf References}

[A] Niels H. Abel, {\it Beweis eines Ausdruckes, von welchem die Binomial-Formel ein einzelner Fall ist},   Crelle's J. Mathematik {\bf 1} (1826), 159-160. \hfill\break
{\tt https://sites.math.rutgers.edu/\~{}zeilberg/akherim/Abel1826.pdf} \quad .

[C] Louis Comtet, {\it ``Advanced Combinatorics''}, D. Reidel Publ. Co.,  Dordrecht/Boston, 1974 \quad . %p. 128

[CKS] F. Chyzak, M. Kauers, B. Salvy,
{\it A non-holonomic systems approach to special function identities}, ISSAC {\bf 2009}. Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation, 111-118.
Association for Computing Machinery (ACM), New York, 2009.

[EM] Shalosh B. Ekhad and John Majewicz, {\it A short WZ-style proof of Abel's identity}, Electronic Journal of Combinatorics {\bf 3(2)} (the Foata  Festschrift) \#R16. \hfill\break
{\tt https://sites.math.rutgers.edu/\~{}zeilberg/mamarim/mamarimPDF/abel.pdf} \quad .

[Fa1] Sister Mary Celine Fasenmyer, {\it Some generalized hypergeometric polynomials}, Bull. Amer. Math. Soc. {\bf 53} (1947), 806-812.

[Fa2]  Sister Mary Celine Fasenmyer,  {\it A note on pure recurrence relations}, Amer. Math. Monthly {\bf 56} (1949), 14-17.

[Fo]  Dominique Foata, {\it Enumerating $k$-Trees}, Discrete Mathematics {\bf 2}  (1971), 181-186.\hfill\break
{\tt https://irma.math.unistra.fr/\~{}foata/paper/pub16.pdf} \quad .

[FoFu]  Dominique Foata and Aim\'e Fuchs, {\it  R\'earrangements de fonctions et d\'enombrement}, Discrete Mathematics {\bf 2}  (1971), 181-186. \hfill\break
{\tt https://irma.math.unistra.fr/\~{}foata/paper/pub12.pdf} \quad .

[Fr] Jean Fran{\c c}on, {\it Preuves combinatoires des identit\'es d'Abel}, Discrete Mathematics  {\bf 8} (1974), 331-343.

[H] Adolf Hurwitz, {\it \"Uber Abel’s Verallgemeinerung der binomischen Formel,} Acta Math. 26 (1902), 199-203.

[Ka] Gil Kalai, {\it A note on an evaluation of Abel sums},  J. Combin, Theory (Ser. A), {\bf 27} (1979), 213-217.\hfill\break
{\tt https://sites.math.rutgers.edu/\~{}zeilberg/akherim/kalai1979.pdf} \quad .


[Ka'] Gil Kalai, Nostalgia corner: John Riordan's referee report of my first paper, blog post on 
Combinatorics and More, 2021. https://gilkalai.wordpress.com/2021/02/19/nostalgia-corner-john-riordans-referee-report-of-my-first-paper/ 

[Kn] Donald E. Knuth, {\it The Art of Computer Programming,} Vol. I: Fundamental Algorithms (Addison-Wesley, Reading, MA, 1969).

[M1]  John Majewicz, {\it WZ-type certification procedures and Sister Celine's technique for  Abel-type sums}, J. of Difference Equations and Applications {\bf 2} (1996),  55-65.

[M2]  John Majewicz, {\it WZ certification of Abel-type identities and Askey's positivity conjecture}, Ph.D. thesis, Temple University 1997. \hfill\break
{\tt https://sites.math.rutgers.edu/\~{}zeilberg/Theses/JohnMajewiczThesis.pdf} \quad .

[PWZ] Marko Petkovsek, Herbert S. Wilf, and Doron Zeilberger, {\it ``A=B''}, A.K. Peters, 1996. \hfill\break
{\tt https://sites.math.rutgers.edu/\~{}zeilberg/AeqB.pdf} \quad .

[R] John Riordan, {\it ``Combinatorial identities''}, Wiley, New York, 1968.

[Z] Doron Zeilberger,  {\it Sister Celine's technique and its generalizations}, 
J. Math. Anal. Appl. {\bf 85} (1982), 114-145. \hfill\break
{\tt https://sites.math.rutgers.edu/\~{}zeilberg/mamarimY/celine1982.pdf} \quad .

\bigskip
\hrule
\bigskip
Gil Kalai, Hebrew University of Jerusalem, Einstein Institute of Mathematics, and
Reichman University, Efi Arazi School of Computer Science. \hfill\break
Email: {\tt gil dot kalai at gmail dot com} \quad .

\bigskip
Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen
Rd., Piscataway, NJ 08854-8019, USA. \hfill\break
Email: {\tt DoronZeil at gmail  dot com}   \quad .
\bigskip
First witten: {\bf May 1, 2024} \quad . This revised version (following the insightful comments by the editor and referees): {\bf Aug. 16, 2024}.

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