A note on a generalization of two well-known Cominatorial identities via a Hypergeometric series approach
aa r X i v : . [ m a t h . C A ] J u l A NOTE ON A GENERALIZATION OF TWO WELL-KNOWNCOMBINATORIAL IDENTITIES VIA A HYPERGEOMETRRICSERIES APPROACH
ARJUN K. RATHIE, INSUK KIM ∗ AND RICHARD B. PARIS
Abstract.
In this note, we aim to provide generalizations of (i) Knuth’s oldsum (or Reed Dawson identity) and (ii) Riordan’s identity using a hypergeo-metric series approach. Introduction and Results required
We start with the following well-known combinatorial sums known as Knuth’sold sums [3], or alternatively as the Reed Dawson identities viz.(1.1) ν X k =0 ( − k (cid:18) νk (cid:19) − k (cid:18) kk (cid:19) = 2 − ν (cid:18) νν (cid:19) and(1.2) ν +1 X k =0 ( − k (cid:18) ν + 1 k (cid:19) − k (cid:18) kk (cid:19) = 0 . It is of interest to mention that Reed Dawson presented the above identities in aprivate communication to Riordan who recorded them in his well-known book [9,pp. 71].Several different proofs of the above sums have been given in the literature; seethe survey paper by Prodinger [6]. Jonassen & Knuth [4] gave an elementary demon-stration using a recursion for the binomial coefficients. Gessel [3] expressed thebinomial coefficients as coefficients in appropriate generating functions. Rousseau[4] showed that the sums could be expressed in terms of the constant coefficient inthe expansion of ( x + x − ) n and Prodinger [7] employed the Euler transformation.In 1974, Andrews [1, pp. 478] established the above sums by employing the Gausssecond summation theorem [8] given by(1.3) F (cid:20) a, b ( a + b + 1); 12 (cid:21) = Γ( ) Γ( a + b + )Γ( a + ) Γ( b + ) . In 2004, Choi et al. [2] utilized the following terminating hypergeometric identitiesrecorded, for example in [8, pp. 126, 127], (1.4) F (cid:20) − n, α α ; 2 (cid:21) = ( ) n ( α + ) n and(1.5) F (cid:20) − n − , α α ; 2 (cid:21) = 0 , where ( α ) n denotes the Pochhammer symbol (or the rising factorial) for any com-plex number α ( = 0) defined by( α ) n = ( α ( α + 1) . . . ( α + n − , ( n ∈ N )1 , ( n = 0) . Also, the following well-known combinatorial identities established by Riordan[9] are seen to be closely related to (1.1) and (1.2) viz.(1.6) ν X k =0 ( − k (cid:18) ν + 1 k + 1 (cid:19) − k (cid:18) kk (cid:19) = 2 − ν (2 ν + 1) (cid:18) νν (cid:19) and(1.7) ν +1 X k =0 ( − k (cid:18) ν + 2 k + 1 (cid:19) − k (cid:18) kk (cid:19) = 2 − ν − ( ν + 1) (cid:18) νν (cid:19) . Riordan [9] established (1.6) and (1.7) by the method of inverse relations.Very recently, generalizations of the hypergeometric identities (1.4) and (1.5)were given by Kim et al. [5], written in the following form: F (cid:20) − n, α α + i ; 2 (cid:21) = 2 − α − i Γ( α ) Γ(1 − α )Γ( α + i ) Γ(1 − α − i )(1.8) × i X r =0 ( − r (cid:18) ir (cid:19) Γ( − α − i + r ) ( + i − r ) n Γ( − i + r ) ( α + + i − r ) n and F (cid:20) − n − , α α + i ; 2 (cid:21) = − − α − i Γ( α )Γ(1 − α )Γ( α + i ) Γ(1 − α − i )(1.9) × i X r =0 ( − r (cid:18) ir (cid:19) Γ( − α − i + r ) (1 + i − r ) n Γ( − i + r ) ( α + 1 + i − r ) n , which are valid for i ∈ N .In this note, we aim to provide generalizations of Knuth’s old sums (or ReedDawson identities) (1.1) and (1.2) and Riordan’s identities (1.6) and (1.7) in themost general form for any i ∈ N . In order to obtain the results in the mostgeneral form for any i ∈ N , we have to construct two master formulas. The resultsare established with the help of (1.8) and (1.9). In Section 3 we present cases ofour general result that correspond to Knuth’s old sums and Riordan’s identities,together with some interesting new results. note on a generalization of two combinatorial identities 3 Generalizations
The generalizations of Knuth’s old sums (or Reed Dawson identities) are givenin the following theorem:
Theorem 2.1.
For i ∈ N , the following results hold true. ν X k =0 ( − k (cid:18) ν + ik + i (cid:19) − k (cid:18) kk (cid:19) (2.1) = π (2 ν + 1) i i i !(2 i )! i X r =0 − r (cid:0) ir (cid:1) ( + ( i − r )) ν ( i − r )! Γ ( + ( r − i )) (1 + ( i − r )) ν and ν +1 X k =0 ( − k (cid:18) ν + 1 + ik + i (cid:19) − k (cid:18) kk (cid:19) (2.2) = 2 π (2 ν + 2) i i i !(2 i )! i X r =0 − r (cid:0) ir (cid:1) (1 + ( i − r )) ν ( i − r + 1)! Γ ( ( r − i )) ( + ( i − r )) ν . Proof.
The proof of the identities (2.1) and (2.2) is straightforward. For this, letus consider the sum for i ∈ N : S = n X k =0 ( − k (cid:18) n + ik + i (cid:19) − k (cid:18) kk (cid:19) . Making use of the identities( n − k )! = ( − k n !( − n ) k , Γ(2 z ) = 2 z − Γ( z ) Γ( z + ) √ π , we have after some simplification, S = Γ( n + 1 + i )Γ(1 + i ) Γ( n + 1) n X k =0 ( − n ) k ( ) k − k (1 + i ) k k !(2.3) = Γ( n + 1 + i )Γ(1 + i ) Γ( n + 1) F (cid:20) − n, i ; 2 (cid:21) . Now for n = 2 ν (even) and n = 2 ν + 1 (odd), the F appearing in (2.3)canbe evaluated with the help of the known results (1.8) and (1.9) and after somesimplification, we easily arrive at the results (2.1) and (2.2) asserted by the theorem.This completes the proof of (2.1) and (2.2). (cid:3) In the next section, we shall mention the known summations presented in Section1 as well as two new cases of our main findings.3.
Corollary
1. If we take i = 0 in Theorem 2.1, we obtain(3.1) ν X k =0 ( − k (cid:18) νk (cid:19) − k (cid:18) kk (cid:19) = ( ) ν (1) ν A. Rathie, I. Kim and R. Paris and(3.2) ν +1 X k =0 ( − k (cid:18) ν + 1 k (cid:19) − k (cid:18) kk (cid:19) = 0 , which are equivalent to Knuth’s old sums (or the Reed Dawson identities) (1.1) and(1.2).2. If we take i = 1 in Theorem 2.1, we obtain(3.3) ν X k =0 ( − k (cid:18) ν + 1 k + 1 (cid:19) − k (cid:18) kk (cid:19) = (2 ν + 1) ( ) ν (1) ν and(3.4) ν +1 X k =0 ( − k (cid:18) ν + 2 k + 1 (cid:19) − k (cid:18) kk (cid:19) = ( ν + 1) ( ) ν (2) ν , which are equivalent to the Riordan identities (1.6) and (1.7).3. Finally, if we take i = 2 and i = 3 in Theorem 2.1, we find the following newresults.(3.5) ν X k =0 ( − k (cid:18) ν + 2 k + 2 (cid:19) − k (cid:18) kk (cid:19) = 13 (4 ν + 3) ( ) ν (1) ν (3.6) ν +1 X k =0 ( − k (cid:18) ν + 3 k + 2 (cid:19) − k (cid:18) kk (cid:19) = 2 ( ) ν (1) ν . and(3.7) ν X k =0 ( − k (cid:18) ν + 3 k + 3 (cid:19) − k (cid:18) kk (cid:19) = 15 (8 ν + 5) ( ) ν (1) ν (3.8) ν +1 X k =0 ( − k (cid:18) ν + 4 k + 3 (cid:19) − k (cid:18) kk (cid:19) = 15 (8 ν + 15) ( ) ν (1) ν . Similarly, other results can be obtained.
Acknowledgement
The research work of Insuk Kim is supported by Wonkwang University in 2020.
Authors’ contributions
All authors contributed equally to writing of this paper. All authors read andapproved the final manuscript.
Competing interest
The authors declare that they have no competing interests.
Authors’ affiliations note on a generalization of two combinatorial identities 5
Arjun K. Rathie: Department of Mathematics, Vedant College of Engineeringand Technology (Rajasthan Technical University), Bundi, 323021, Rajasthan, IndiaE-mail : [email protected] Kim: Department of Mathematics Education, Wonkwang University, Ik-san, 570-749, Republic of KoreaE-mail: [email protected] B. Paris: Division of Computing and Mathematics, University of Aber-tay, Dundee DD1 1HG, UKE-mail: [email protected]
References [1] Andrews, G.E.,
Applications of Basic Hypergeometric Functions , SIAM Review, 16(4), 441-484, (1974).[2] Choi, J., Rathie, A.K. and Harsh, H.V.,
A note on Reed Dawson identities , Korean J. Math.Sciences, 11(2), 1-4, (2004).[3] Greene, D. and Knuth, D.E.,
Mathematics for the Analysis of Algorithms , Birkh¨auser, (1981).[4] Jonassen, A. and Knuth, D.E.,
A trivial algorithm whose analysis isn’t , Journal of Computerand System Sciences, 16, 301-322, (1978).[5] Kim, Y.S., Rathie, A.K. and Paris, R.B.
Evaluations of some terminating F (2) series withapplications , Turkish J. Math., 42(5), 2563-2575, (2018).[6] Prodinger, H., Knuth’s old sum-A survey , EATCS Bulletin, 52, 232-245, (1994).[7] Prodinger, H.,
Some information about the binomial transform , The Fibonacci Quarterly,32(5), 412-415, (1994).[8] Rainville, E.D.,
Special Functions , Macmillan Company, New York, (1960); Reprinted byChelsea Publishing Company, Bronx, New York, (1971).[9] Riordan, J.,