aa r X i v : . [ m a t h . C A ] M a y ALTERNATING CONVOLUTIONS OF CATALAN NUMBERS
WENCHANG CHU
Abstract.
A new class of alternating convolutions concerning binomial coef-ficients and Catalan numbers are evaluated in closed forms. Introduction and Motivation
The Catalan numbers C n = 1 n + 1 (cid:18) nn (cid:19) for n ∈ N are probably the most frequently encountered sequence in mathematics. There existnumerous interpretations in enumerative combinatorics and remarkable identitiesabout them that can be found in the monographs by Koshy [6], Roman [10] andStanley [12] as well as in [4,5,13]. For instance, these numbers satisfy the nonlinearrecurrence relation C n +1 = n X k =0 C k C n − k and the Touchard identity C n +1 = ⌊ n ⌋ X k =0 n − k (cid:18) n k (cid:19) C k . Here and forth, ⌊ x ⌋ denotes the integer part for the real number x . For m ∈ N and i, j ∈ Z , we shall utilize the notation “ i ≡ m j ” for “ i is congruent to j modulo m ”. The logical function χ is defined for brevity by χ (true) = 1 and χ (false) = 0.Recently, Miki´c [7, 2019] found, by combinatorial bijections, the following unusualconvolution identities: n X k =0 ( − k (cid:18) nk (cid:19) C k C n − k = 2 χ ( n ≡ n + 2 (cid:18) n ⌊ n ⌋ (cid:19) , (1) n X k =0 ( − k (cid:18) nk (cid:19)(cid:18) n − kn − k (cid:19) C k = (cid:18) n ⌊ n ⌋ (cid:19) . (2)Prodinger [8] provided different proofs by making use of Zeilberger’s algorithm,Dixon’s formula and its variants. Let ( x ) n be the Pochhammer symbol given by( x ) = 1 and ( x ) n = x ( x + 1) · · · ( x + n −
1) for n ∈ N . The objective of this paper is to show the following generalizations.
Mathematics Subject Classification.
Primary 05A10, Secondary 33C15.
Key words and phrases.
Alternating convolution; Catalan number; Binomial coefficient; Hyper-geometric series; Product formula of confluent hypergeometric series.Email address: [email protected].
Wenchang Chu
Theorem 1 ( n, λ ∈ N ) . n X k =0 ( − k (cid:18) nk (cid:19) C k + λ C n − k + λ = λ ! χ ( n ≡ n ) λ (cid:18) λλ (cid:19)(cid:18) n ⌊ n ⌋ (cid:19) C λ + ⌊ n ⌋ . Theorem 2 ( n, λ ∈ N ) . n X k =0 ( − k (cid:18) nk (cid:19)(cid:18) n − k + 2 λn − k + λ (cid:19) C k + λ = λ !(2 + n ) λ (cid:18) λλ (cid:19)(cid:18) n ⌊ n ⌋ (cid:19)(cid:18) n + 2 λλ + ⌊ n ⌋ (cid:19) . Three further binomial identities of alternating convolutions will also be established.
Theorem 3 ( n, λ ∈ N ) . n X k =0 ( − k (cid:18) nk (cid:19)(cid:18) k + 2 λk + λ (cid:19)(cid:18) n − k + 2 λn − k + λ (cid:19) = λ ! χ ( n ≡ n ) λ (cid:18) λλ (cid:19)(cid:18) n ⌊ n ⌋ (cid:19)(cid:18) λ + nλ + ⌊ n ⌋ (cid:19) . Theorem 4 ( n, λ ∈ N ) . n X k =0 ( − k (cid:18) nk (cid:19)(cid:18) k + 2 λk + λ (cid:19)(cid:18) n − k + 2 λn − k + λ (cid:19) ( n − k + λ )= λ !( n ) λ (cid:18) n ⌊ n ⌋ (cid:19)(cid:18) λλ (cid:19)(cid:18) λ + nλ + ⌊ n ⌋ (cid:19) × n (2 λ + n )2( λ + n ) , n ≡ ( n +1)(2 λ + n +1)2( λ + n ) , n ≡ . Theorem 5 ( n, λ, µ ∈ N ) . n X k =0 ( − k (cid:18) nk (cid:19) (cid:18) k + 2 λk + λ (cid:19)(cid:18) n − k + 2 µn − k + µ (cid:19)(cid:18) k + 2 λλ (cid:19)(cid:18) n − k + 2 µµ (cid:19) = (cid:18) n n (cid:19)(cid:18) n + λ + µ n (cid:19)(cid:18) λ + n λ (cid:19)(cid:18) µ + n µ (cid:19) χ ( n ≡ . The rest of the paper will be organized as follows. As a preliminary, we shallprove, in the next section, a product formula for confluent hypergeometric F and F -series, which may serve as a counterpart of the product formulae due to Bai-ley [1]. By extracting the coefficients of x n across these hypergeometric equations,we derive, in Section 3, three binomial formulae of alternating sums, which containthe five summation theorems just displayed as particular cases. Finally, the paperwill end with Section 4, where two equivalent integral formulae are proposed asproblems. 2. Products of Hypergeometric Series
Recall that the Γ-function (see [9, §
8] for example) is given by the beta integralΓ( x ) = Z ∞ u x − e − u d u for ℜ ( x ) > . It satisfies Euler’s reflection propertyΓ( x ) × Γ(1 − x ) = π sin πx lternating Convolutions of Catalan Numbers 3 and Legendre’s duplicate formulaΓ(2 x ) = Γ( x )Γ( x + ) 2 x − √ π . For the sake of brevity, we shall utilize the following multiparameter expressionΓ (cid:20) α, β, · · · , γA, B, · · · , C (cid:21) = Γ( α ) n Γ( β ) n · · · Γ( γ ) n Γ( A ) n Γ( B ) n · · · Γ( C ) n . Analogously, the quotient of the Pochhammer symbol will be abbreviated to (cid:20) α, β, · · · , γA, B, · · · , C (cid:21) n = ( α ) n ( β ) n · · · ( γ ) n ( A ) n ( B ) n · · · ( C ) n . According to Bailey [2, § p F q (cid:20) a , a , · · · , a p b , b , · · · , b q (cid:12)(cid:12)(cid:12) z (cid:21) = ∞ X k =0 z k k ! (cid:20) a , a , · · · , a p b , b , · · · , b q (cid:21) k . When p ≤ q , this series is said confluent. In 1928, Bailey [1] found, among others,two product formulae for confluent hypergeometric series F (cid:20) ac (cid:12)(cid:12)(cid:12) x (cid:21) × F (cid:20) ac (cid:12)(cid:12)(cid:12) − x (cid:21) = F (cid:20) a, c − ac, c , c (cid:12)(cid:12)(cid:12) x (cid:21) , (3) F (cid:20) a a (cid:12)(cid:12)(cid:12) x (cid:21) × F (cid:20) c c (cid:12)(cid:12)(cid:12) − x (cid:21) = F (cid:20) a + c , a + c +12 a + c, a + , c + (cid:12)(cid:12)(cid:12) x (cid:21) . (4)They resemble the following beautiful product formula (cf. Bailey [2, § F (cid:20) a, ca + c + (cid:12)(cid:12)(cid:12) x (cid:21) = F (cid:20) a + c, a, ca + c + , a + 2 c (cid:12)(cid:12)(cid:12) x (cid:21) . By inserting an extra linear factor λ + k in (3), we find the extended formula. Lemma 6 ( a, c, λ ∈ R ) . F (cid:20) ac (cid:12)(cid:12)(cid:12) x (cid:21) × F (cid:20) λ, aλ, c (cid:12)(cid:12)(cid:12) − x (cid:21) = F (cid:20) λ, a, c − aλ, c, c , c (cid:12)(cid:12)(cid:12) x (cid:21) − axcλ F (cid:20) a, c − ac, c , c (cid:12)(cid:12)(cid:12) x (cid:21) . In particular for λ = c −
1, we get a variant formula of (3): F (cid:20) ac − (cid:12)(cid:12)(cid:12) x (cid:21) × F (cid:20) ac (cid:12)(cid:12)(cid:12) − x (cid:21) = F (cid:20) a, c − ac − , c , c (cid:12)(cid:12)(cid:12) x (cid:21) − axc ( c − F (cid:20) a, c − ac, c , c (cid:12)(cid:12)(cid:12) x (cid:21) . (5) Proof of Lemma 6 . By means of the linear relation λ + kk = λ − aλ + a + kλ Wenchang Chu it is not hard to get the contiguous relation F (cid:20) a, c, e, λ a − c, a − e, λ (cid:12)(cid:12)(cid:12) (cid:21) = λ − aλ F (cid:20) a, c, e a − c, a − e (cid:12)(cid:12)(cid:12) (cid:21) × aλ F (cid:20) a, c, e a − c, a − e (cid:12)(cid:12)(cid:12) (cid:21) , where the condition ℜ ( a − c − e ) > F -series by the Dixon formula (cf. Bailey [2, § F (cid:20) a, c, e a − c, a − e (cid:12)(cid:12)(cid:12) (cid:21) = Γ (cid:20) a − c, a − e, a , a − b − c a − c, a − e, a, a − c − e (cid:21) and the latter F -series by “ D − , − ” due to the author [3, Example 18] F (cid:20) a, c, e a − c, a − e (cid:12)(cid:12)(cid:12) (cid:21) = 2 a − c − e − π Γ (cid:20) a − c, a − e a − c, a − e (cid:21) × (cid:26) Γ (cid:20) a , a − c, a − e, a − c − e a, , a − c − e (cid:21) +Γ (cid:20) a , a − c, a − e, a − c − e a, , a − c − e (cid:21) (cid:27) and then simplifying the result, we get the expression F (cid:20) a, c, e, λ a − c, a − e, λ (cid:12)(cid:12)(cid:12) (cid:21) = Γ (cid:20) a − c, a − ea, a − c − e (cid:21) × (cid:26) λ Γ (cid:20) a , a − c − e a − c, a − e (cid:21) + 2 λ − a λ Γ (cid:20) a , a − c − e a − c, a − e (cid:21) (cid:27) . (6)In particular, when the series is terminated by a = − n with n ∈ N , we have F (cid:20) − n, c, e, λ − c − n, − e − n, λ (cid:12)(cid:12)(cid:12) (cid:21) = (cid:20) − n, − c − e − n − c − n, − e − n (cid:21) ⌊ n ⌋× λ + n λ , n ≡ − n λ , n ≡ . (7)Now we turn to examine, by letting i + k = n , the product F (cid:20) ac (cid:12)(cid:12)(cid:12) x (cid:21) × F (cid:20) λ, aλ, c (cid:12)(cid:12)(cid:12) − x (cid:21) = ∞ X i =0 ( a ) i i !( c ) i ∞ X k =0 ( − k λ + kλ ( a ) k k !( c ) k x k + i = ∞ X n =0 ( a ) n x n n !( c ) n n X k =0 (cid:20) − n, a, − c − n, λ + 11 , c, − a − n, λ (cid:21) k . Writing the last sum in terms of F -series and then evaluating it by (7) F (cid:20) − n, a, − c − n, λ + 1 c, − a − n, λ (cid:12)(cid:12)(cid:12) (cid:21) = (cid:20) − n, c − ac, − a − n (cid:21) ⌊ n ⌋ × λ + n λ , n ≡ − n λ , n ≡ n , the product formula in Lemma 6. (cid:3) Binomial Convolution Formulae
By extracting the coefficient of x n across the equations (3), (4) and (5) on hyper-geometric products, we find the following three identities. lternating Convolutions of Catalan Numbers 5 Proposition 7 ( n ∈ N and a, c ∈ R ) . n X k =0 ( − k (cid:18) nk (cid:19) ( a ) k ( a ) n − k ( c ) k ( c ) n − k = n !( c ) n (cid:20) a, c − a , c (cid:21) ⌊ n ⌋ χ ( n ≡ . Proposition 8 ( n ∈ N and a, c ∈ R ) . n X k =0 ( − k (cid:18) nk (cid:19) ( a ) k ( c ) n − k (2 a ) k (2 c ) n − k = (cid:20) , a + c a, c (cid:21) n (cid:20) a, c , a + c (cid:21) ⌊ n ⌋ χ ( n ≡ . Proposition 9 ( n ∈ N and a, c ∈ R ) . n X k =0 ( − k (cid:18) nk (cid:19) ( a ) k ( a ) n − k ( c ) k ( c − n − k = n !( c − n +1 (cid:20) a, c − a , c (cid:21) ⌊ n ⌋ × c + n − , n ≡ a + n − , n ≡ . Expressing the quotients of shifted factorials in terms of binomial coefficients( + λ ) k (1 + λ ) k = (cid:0) k +2 λk + λ (cid:1) k (cid:0) λλ (cid:1) , ( + λ ) k (1 + λ ) k = (cid:0) k +2 λk + λ (cid:1) (1 + 2 k + 2 λ )4 k (cid:0) λλ (cid:1) (1 + 2 λ ) ;( + λ ) k (2 + λ ) k = C k + λ k C λ , ( + λ ) k (2 + λ ) k = (cid:0) k +2 λk + λ (cid:1) k (cid:0) λλ (cid:1) ;we can confirm the five identities anticipated in the introduction as follows: • Theorem 1: a = + λ and c = 2 + λ in Proposition 7. • Theorem 2: a = + λ and c = 2 + λ in Proposition 9. • Theorem 3: a = + λ and c = 1 + λ in Proposition 7. • Theorem 4: a = + λ and c = 1 + µ in Proposition 9. • Theorem 5: a = + λ and c = 2 + µ in Proposition 8.Furthermore, we can derive four “reciprocal formulae” of those displayed in Theo-rems 1, 2, 3 and 4. Corollary 10 ( a = 2 + λ and c = + λ in Theorem 7: n, λ ∈ N ) . n X k =0 ( − k (cid:18) nk (cid:19) ( n − n − C λ C k + λ C n − k + λ = 3 C λ (cid:0) λλ (cid:1)(cid:0) n ⌊ n ⌋ (cid:1) χ ( n ≡ C λ + ⌊ n ⌋ (cid:0) λ + nλ (cid:1)(cid:0) λ +2 nλ + n (cid:1) . Corollary 11 ( a = 2 + λ and c = + λ in Theorem 9: n, λ ∈ N ) . n X k =0 ( − k (cid:18) nk (cid:19) n (cid:0) λλ (cid:1) C λ (cid:0) k +2 λk + λ (cid:1) C n − k + λ = (1 + n + 2 λ ) C λ (cid:0) λλ (cid:1)(cid:0) n ⌊ n ⌋ (cid:1)(cid:0) λ + n +1 n (cid:1)(cid:0) λ +2 nλ + n (cid:1)(cid:0) λ + nλ + ⌊ n ⌋ (cid:1) × n − n , n ≡ n − n , n ≡ . Corollary 12 ( a = 1 + λ and c = + λ in Theorem 7: n, λ ∈ N ) . n X k =0 ( − k (cid:18) nk (cid:19) (1 − n ) (cid:0) λλ (cid:1) (cid:0) k +2 λk + λ (cid:1)(cid:0) n − k +2 λn − k + λ (cid:1) = (cid:0) λλ (cid:1) (cid:0) n ⌊ n ⌋ (cid:1) χ ( n ≡ (cid:0) λ + nn (cid:1)(cid:0) λ +2 nλ + n (cid:1)(cid:0) λ + nλ + ⌊ n ⌋ (cid:1) . Wenchang Chu
Corollary 13 ( a = 1 + λ and c = + λ in Theorem 9: n, λ ∈ N ) . n X k =0 ( − k (cid:18) nk (cid:19) n (1 + 2 λ )(1 + 2 n + 2 λ ) (cid:0) λλ (cid:1) (1 + 2 k + 2 λ ) (cid:0) k +2 λk + λ (cid:1)(cid:0) n − k +2 λn − k + λ (cid:1) = (1 + 2 λ ) (cid:0) λλ (cid:1) (cid:0) n ⌊ n ⌋ (cid:1)(cid:0) λ + nn (cid:1)(cid:0) λ +2 nλ + n (cid:1)(cid:0) λ + nλ + ⌊ n ⌋ (cid:1) × n, n ≡ n + 1 , n ≡ . Integral representations
According to the expressions of the beta integrals (cf. [11]) (cid:18) mm (cid:19) = 2 m π β ( , + m ) = 2 m π Z x m − √ − x d x,C m = 2 m π β ( , + m ) = 2 m π Z y m − p − y d y ;we can reformulate the sum in Theorem 1 as follows n X k =0 ( − k (cid:18) nk (cid:19) C k + λ C n − k + λ = 4 n +2 λ π Z Z ( xy ) λ − p (1 − x )(1 − y ) n X k =0 ( − k (cid:18) nk (cid:19) x n − k y k d x d y = 4 n +2 λ π Z Z ( xy ) λ − ( x − y ) n p (1 − x )(1 − y )d x d y. Then we get the following integral formula equivalent to Theorem 1.
Corollary 14 ( n, λ ∈ N ) . Z Z ( xy ) λ − ( x − y ) n p (1 − x )(1 − y ) d x d y = π λ ! χ ( n ≡ n +2 λ (2 + n ) λ (cid:18) λλ (cid:19)(cid:18) n ⌊ n ⌋ (cid:19) C λ + ⌊ n ⌋ . The sum in Theorem 2 can analogously be manipulated: n X k =0 ( − k (cid:18) nk (cid:19)(cid:18) n − k + 2 λn − k + λ (cid:19) C k + λ = 2 n +4 λ π Z Z ( xy ) λ − r − y − x n X k =0 ( − k (cid:18) nk (cid:19) x n − k y k d x d y = 2 n +4 λ π Z Z ( xy ) λ − ( x − y ) n r − y − x d x d y, which leads us to another integral formula. Corollary 15 ( n, λ ∈ N ) . Z Z ( xy ) λ − ( x − y ) n r − y − x d x d y = π λ !2 n +4 λ (2 + n ) λ (cid:18) λλ (cid:19)(cid:18) n ⌊ n ⌋ (cid:19)(cid:18) n + 2 λλ + ⌊ n ⌋ (cid:19) . Two further integral formulae corresponding to Theorems 3 and 4 can be producedin a similar manner. Finally, an intriguing question is how to evaluate these integralsdirectly? lternating Convolutions of Catalan Numbers 7
References [1] W. N. Bailey,
Products of generalized hypergeometric series,
Proc. London Math. Soc. (2) 28(1928), 242–254.[2] W. N. Bailey,
Generalized Hypergeometric Series,
Cambridge University Press, Cambridge,1935.[3] W. Chu,
Analytical formulae for extended F -series of Watson–Whipple–Dixon with twoextra interger parameters, Math. Comp. 81:277 (2012), 467–479.[4] W. Chu,
Lattice paths and the q -ballot polynomials, Adv. Appl. Math. 87 (2017), 108–127.[5] W. Chu,
Further identities on Catalan numbers,
Discrete Math. 341:11 (2018), 3159–3164.[6] T. Koshy,
Catalan Numbers with Applications,
Oxford University Press, New York, 2009.[7] J. Miki´c,
Two new identities involving the Catalan numbers and sign–reversing involutions,
J.Integer Seq. 19 (2019), Art
Two new identities involving the Catalan numbers: A classical ap-proach, arXiv:1911.07604v1 [math.CO] 18 Nov 2019.[9] E. D. Rainville,
Special Functions,
New York, The Macmillan Company, 1960.[10] S. Roman,
An Introduction to Catalan Numbers,
Birkh¨auser, Heidelberg, 2015.[11] R. Sprugnoli,
Sums of reciprocals of the central binomial coefficients,
INTEGERS 6 (2006),
Catalan Numbers,
Cambridge University Press, Cambridge, 2015.[13] R. R. Zhou and W. Chu,
Identities on extended Catalan numbers and their q -analogs, GraphsCombin. 32:5 (2016), 2183–2197.
School of Mathematics and StatisticsZhoukou Normal UniversityZhoukou (Henan), P. R. China
Corresponding address :Department of Mathematics and PhysicsUniversity of Salento (P. O. Box 193)73100 Lecce, Italy
E-mail address ::