New formulas for Stirling-like numbers and Dobinski-like formulas

Lucky 13-th Exercises on S tirling-like numbers an d Do binski-like f ormulas Andrzej Krzysztof Kwa ´ sniewski Member of the Inst itute of Comb inatorics and its Applications High School of M athematics and Applied Informatics Kamienna 17, PL-15-021 Bia lystok, Poland e-mail: kwand r@gmail.com Summary Extensions of the Stirling num b ers of the second kind and Dobinski-lik e formulas are prop osed in a series of ex ercises for graduetes. Some of these new formulas recently disco vered b y me are to b e found in A.K.Kw a ´ sniewski’s source pap er [1]. These exten- sions naturally encompass th e w ell known q -extensions.The indicatory references are to p oint at a p art of th e vas t domain of th e foundations of computer science in ArX iv affiliation noted as CO.cs .DM. MCS n umb ers: 05A40, 11B7 3, 81S99 Keywo rds: umbral calculus, extended Stirling num bers, D obinski typ e identities affiliated t o The Internet Gian-Carlo Polish S emin ar: http://ii.uwb.e du.pl /akk/sem/sem r ota.htm Published i n: Proc. Jang jeon M ath. So c. V ol. 11 No 2 , ( 2008 ),137-144 1. In the q -extensions realm Ex.1 Recall and pro ve it again o ccasionally n oting that The N umber 44 is a magic num b er in Poli sh Poetry and this had had an imp lemen tation in quite recent P olish history in 1968 s tudents revolutionary riots for indep endence and freedom of th inking under comm un ist regime. W ell, then f ort y f our y ears ago Gi an-Carlo Rota [4] pro ved that the exp onential generating function for Bell n umbers B n is of t he form ∞ X n =0 x n n ! B n = exp( e x − 1) (1) using th e linear functional L such that L ( X n ) = 1 , n ≥ 0 (2) Then Bell n umbers (see: form ula (4) in [4]) a re defi ned by L ( X n ) = B n , n ≥ 0 (3) The abov e formula is exactly the Dobinski form ula [5] if L is interpreted as the a vera ge v alue functional for the random v ariable X with the P oisson distribution with L ( X ) = 1. Recall it and prov e i t again. Ex.2 Recalland pro ve : The t w o standard [12],see also [13-17] q -extensions Stirling numbers of the second kind are defined by x n q = n X k =0  n k ff q x k q , (4) 1 where x q = 1 − q x 1 − q and x k q = x q ( x − 1) q ... ( x − k + 1) q , whic h corre sp onds to the ψ sequence choice in the q - Gauss form h 1 n q ! i n ≥ 0 and q ∼ -Stirling numbers x n = n X k =0  n k ff ∼ q χ k ( x ) (5) where χ k ( x ) = x ( x − 1 q )( x − 2 q ) ... ( x − [ k − 1] q ) Note that th ese formulae b ecome the usual unexten ded Stirling numbers of the second kind formulae when the subscript q is remov ed . Ex.3 Recall and prov e it again: F or these tw o classical by now q - extensions of Stirling numbers of th e second kind - the ” q -stand ard” recurrences hold resp ectively:  n + 1 k ff q = n X l =0 n l ! q q l  l k − 1 ff q ; n ≥ 0 , k ≥ 1 ,  n + 1 k ff ∼ q = n X l =0 n l ! q q l − k +1  l k − 1 ff ∼ q ; n ≥ 0 , k ≥ 1 . Show that these form ulae b ecome th e usual unextend ed Stirling num b ers of the second kind formula e when th e subscript q is re mov ed and th e number q is p ut equal to one. Ex.4 Recall and prov e it again: F rom the above it follo ws immediately th at corresp onding q - extensions of B n Bell num b ers s atisfy res p ective recurrences: B q ( n + 1) = n X l =0 n l ! q q l B q ( l ); n ≥ 0 , B ∼ q ( n + 1) = n X l =0 n l ! q q l − k +1 B ∼ q ( l ); n ≥ 0 where B ∼ q ( l ) = l X k =0 q k  l k ff ∼ q . Show that these form ulae b ecome th e usual unextend ed Stirling num b ers of the second kind formula e when th e subscript q is re mov ed and th e number q is p ut equal to one. Ex.5 Recall and prov e it again: Recursions for b oth i nversion q -Bell num b ers and inve rsion q -Stirling num b ers of the second kind are not difficult to b e d erived. Also in a natural wa y the inversion q - Stirling numbers of the second kind from [16] satisfy a q -analogue of th e standard recursion f or Stirling num b ers of the second kind to be w ritten via mnemonic adding ” q ” subscript to the b in omial and second kind Stirling symb ols in th e the standard recursion formula i.e.  n + 1 k ff inv q = n X l =0 n l ! q  n − l k − 1 ff inv q ; n ≥ 0 , k ≥ 1 . Another q -ex t ended Stirling numbers muc h d ifferent from Carlitz ” q -ones” were intro- duced in the reference [19],see [1,2 0,28]. The cigl - q -Stirling num b ers of the second kind are exp ressed in terms of q -binomial coefficients and q = 1 St irling num b ers of the second kind [16,17],( see [1] for more references) as fol lo ws 2  n + 1 k ff cigl q = n X l =0 n l ! q q ( n − l +1 2 )  n − l k − 1 ff cigl q ; n ≥ 0 , k ≥ 1 . The corresp onding cigl - q - Bell num b ers recen tly hav e b een eq uiv alently d efined via cigl - q - Dobinski form ula [ 20,28] - whic h now in mor e adeq uate n otation reads : L ( X q n ) = B n ( q ) , n ≥ 0 , X q n ≡ X ( X + q − 1) ... ( X − 1 + q n − 1 ) . The ab ov e c igl - q -Dobinski formula is interpreted as the av erage of this sp ecific n − th cigl - q - p o w er random v ariable X q n with the q = 1 Po isson distribution such that L ( X ) = 1. F or that to see use t h e identity by Cigler [19] x ( x − 1 + q ) ... ( x − 1 + q n − 1 ) = n X k =0  n k ff cigl q x k . Note that these form ulae b ecome the usual unextend ed Stirling num b ers of the second kind formulae when the subscript q is remove d and the number q is put eq u al to one. 2. Bey ond the q -extensions r ealm The further consecutive um bral exten sion o f Carli tz-Gould q - Stirling n u mbers  n k ff q and  n k ff ∼ q is re alized t wo-fol d w ay - one of which leads to a surprise ( ?) in con trary p erhaps t o the other w ay . The first ”easy wa y ” consists in almost mn emonic sometimes replacement of q subscript by ψ after having realized that via equation (5) we are dealing with the sp ecific case of Com tet n umbers [1] i.e. n ow we hav e x n = n X k =0  n k ff ∼ ψ ψ k ( x ) (6 ) where ψ k ( x ) = x ( x − 1 ψ )( x − 2 ψ ) ... ( x − [ k − 1] ψ ) . Show th at th ese formulae b ecome the usual un extended Stirling n umbers of the second kin d form ulae when the subscript ψ is re mov ed. As a consequence we hav e ”for granted” the f ollo wing: Ex.6 Recall standard and prov e its extension:  n + 1 k ff ∼ ψ =  n k − 1 ff ∼ ψ + k ψ  n k ff ∼ ψ ; n ≥ 0 , k ≥ 1; (7) where  n 0 ff ∼ ψ = δ n, 0 ,  n k ff ∼ ψ = 0 , k > n ; and the recurrence for ordinary generating f unction rea ds G ∼ k ψ ( x ) = x 1 − k ψ G ∼ k ψ − 1 ( x ) , k ≥ 1 (8) where naturall y G ∼ k ψ ( x ) = X n ≥ 0  n k ff ∼ ψ x n , k ≥ 1 from where one infers 3 G ∼ k ψ ( x ) = x k (1 − 1 ψ x )(1 − 2 ψ x ) ... (1 − k ψ x ) , k ≥ 0 (9) hence w e a rrive in the stand ard extended tex t-b o ok wa y [21] at the follo wing ex plicit form ula  n k ff ∼ ψ = r n ψ k ψ ! k X r =1 ( − 1) k − r k ψ r ψ ! ; n, k ≥ 0 . (10) Show that these form ulae b ecome th e usual unextend ed Stirling num b ers of the second kind formulae when the subscript ψ is remov ed. Expanding th e right h and side of the corresp onding equation ab ov e results in another explicit formula for these ψ -case Com tet n umb ers [1] i .e. we hav e Ex.7 Recall standard and pro ve its extension:  n k ff ∼ ψ = X 1 ≤ i 1 ≤ i 2 ≤ ... ≤ i n − k ≤ k ( i 1 ) ψ ( i 2 ) ψ ... ( i n − k ) ψ ; n, k ≥ 0 . (11) or equ iva lently (compare with [13], see [12 ,14,15] )  n k ff ∼ ψ = X d 1 + d 2 + ... + d k = n − k, d i ≥ 0 1 d 1 ψ 2 d 2 ψ ...k d k ψ ; n, k ≥ 0 . (12) ψ ∼ - Stirling num b e rs of t h e second kind b eing defined equ iv alently by (10 ) , ( 14), (15) or (16) yield ψ ∼ - Bell num b ers B ∼ n ( ψ ) = n X k =0  n k ff ∼ ψ , n ≥ 0 . Show that these form ulae b ecome th e usual unextend ed Stirling num b ers of the second kind formulae when the subscript ψ is remov ed. Ex.8 Recall standard and pro ve its extension: Adapting the stand ard text-b ook method [2 1] w e ha ve for t w o va riable o rdinary g en- erating function for  n k ff ∼ ψ Stirling num b ers of the second kind and the ψ -exp onen t ial generating f unction for B ∼ n ( ψ ) Bell n umbers the f ollo wing formula e C ∼ ψ ( x, y ) = X n ≥ 0 A ∼ n ( ψ , y ) x n , (13) where the ψ - exponential-lik e polyn omials A ∼ n ( ψ , y ) A ∼ n ( ψ , y ) = n X k =0  n k ff ∼ ψ y k do satisfy the recurrence A ∼ n ( ψ , y ) = [ y (1 + ∂ ψ ] A ∼ n − 1 ( ψ , y ) n ≥ 1 , hence A ∼ n ( ψ , y ) = [ y (1 + ∂ ψ ] n 1 , n ≥ 0 , where the linear op erator ∂ ψ acting on the algebra of formal p ow er series is b eing called (see: [1,2,3,2 4,25,31 ]) the ” ψ -deriva tive” and ∂ ψ y n = n ψ y n − 1 . Show that these form ulae b ecome th e usual unextend ed Stirling num b ers of the second kind formulae when the subscript ψ is remov ed. 4 Ex.9 Recall standard and prov e its extension: The ψ -exponential generating f unction B ∼ ψ ( x ) = P n ≥ 0 B ∼ n ( ψ ) x n n ψ ! for B ∼ n ( ψ ) Bell numbers - after cautious adaptation of the meth o d from the Wilf‘s generatingfunctionology bo ok [ 21] can be seen t o be giv en b y the follow ing form ula B ∼ ψ ( x ) = X r ≥ 0 1 ǫ ( ψ , r ) e ψ [ r ψ x ] r ψ ! (14) where (see : [1,2,3,24, 25,31]) e ψ ( x ) = X n ≥ 0 x n n ψ ! while ǫ ( ψ , r ) = ∞ X k = r ( − 1) k − r ( k ψ − r ψ )! (15) and f or the ψ -extensions the Dobinski like formula here no w reads B ∼ n ( ψ ) = X r ≥ 0 1 ǫ ( ψ , r ) r n ψ r ψ ! . (16) Show th at these formulae b ecome the usual unextended Stirling numbers of the second kind formulae when the subscript ψ is remo ved. Ex.10 R ecall standard and prov e its ext en sion: In the case of Gauss q -extend ed choice of h 1 n q ! i n ≥ 0 admissible sequence of extended umbral operator calculus equations (19) and (20 ) t ake the form ǫ ( q , r ) = ∞ X k = r ( − 1) k − r ( k − r ) q ! q − ( r 2 ) (17) and the q ∼ -Dobinski formula is given by B ∼ n ( q ) = X r ≥ 0 1 ǫ ( q , r ) r n q r q ! , (18) whic h for q = 1 b ecomes the Dobin sk i form ula from 1887 [5]. Ex.11 Recall standard and prov e its exten sion: In a du al inverse wa y w e define the ψ ∼ Stirling numbers of the first kind as coefficients in th e follo wing exp ansion ψ k ( x ) = k X r =0 » k r – ∼ ψ x r (19) where - recall ψ k ( x ) = x ( x − 1 ψ )( x − 2 ψ ) ... ( x − [ k − 1] ψ ); ( attention: see equ ations (10)-(16) in [7,8] and note the difference with the p resen t defin ition). Therefore from the ab o ve w e infer that k X r =0 » k r – ∼ ψ  r l ff ∼ ψ = δ k,l . (20) Show th at these formulae b ecome the usual unextended Stirling numbers of the second kind formulae when the subscript ψ is remov ed . Ex.12 Recall standard and prov e its exten sion: Consider n o w another Stirling-like num b ers (as exp ected Whitney num bers [31,1,3 2]) whic h are natural counterpart to ψ ∼ -Stirling num b ers of the second kind. These are ψ c - Stirling num b ers of the first kin d defined here down as co efficien ts in t he follo wing expansion ( upp erscript ” c ” is used b ecause of cycles in n on-extended case). 5 ψ k ( x ) = k X r =0 » k r – c ψ x r (21) where - now ψ k ( x ) = x ( x + 1 ψ )( x + 2 ψ ) ... ( x + [ k − 1] ψ ); Show that these form ulae b ecome th e usual unextend ed Stirling num b ers of the second kind formulae when the subscript ψ is remov ed . Ex.13 Recall standard and pro ve its extension: Show that the definition here up stairs ab ov e of the q -Stirling numbers of the second kind  n k ff q is equiva lent with the d efinition by recursion  n + 1 k ff q = q k − 1  n k − 1 ff q + k q  n k ff q ; n ≥ 0 , k ≥ 1; (22) where  n 0 ff q = δ n, 0 ,  n k ff q = 0 , k > n Show that these in turn (just u se th e Q -Leibn itz rule [2,3, 24,25,31 ] for Jac kson deriv ative ∂ q ) are equiv alen t to ( ˆ x∂ q ) n = n X k =0  n k ff q ˆ x k ∂ k q (23) where  n 0 ff q = δ n, 0 ,  n k ff q = 0 , k > n . Here ˆ x denotes the multiplication by t he argument of a function. Show that these form ulae b ecome th e usual unextend ed Stirling num b ers of the second kind formulae when the subscript q is remov ed. Consult [33-35] for some new op en problems arising in related the domain of the so called cob ewb p osets and their acyclic digraphs representative s which had served the present author t o discov er a joint combinatori al interpretation for all F -nomial co ef- ficients. These family encompasses bin omial and q -Gaussian coefficient, Fib onomial coefficient, Stirling numbers of b oth kinds and all classical F − nomial co efficients hence sp ecifically in ciden ce co efficients of reduced incidence algebras of full binomial type and W h itney numbers are given th e joint cobw eb combinatorial interpretation also [36,37]. Ackno wle dgements The aut h or appraises muc h Maciej Dziemia´ n czuk Gda ´ nsk Uni- verit y Student’s assistance includ ing T eX − nolog y aid. The author exp resses his gratitude also Dr Ewa Krot-Sieniaw sk a for her several years ’ co op eration and vivid application of the alike material deserving Stud ents’ ad- miration for her b eing such a comprehensible and reliable T eac her b efore she was fired by Bial ystok Universit y authorities exactly on the day she had de- fended Rota and cobw e b p osets related disse rtation w i th disti nction . References [1] A. Krzysztof Kw a’sniewski On umbr al extensions of Stirling numb ers and Dobinski-like f ormulas Ad v.Stud.Contemp. Math. , V ol 12 , no. 1, ( 2006 )73-100 cs.DM ArXiv : math.CO/0411002 [2] A. Krzy sztof Kwa’snie wski Main the or ems of exte nde d finite op er ator c alculus Integral T ransforms and Sp ecial F u nctions, 14 No 6 (2003): 499-516 [3] A . Krzysztof Kwa ’sniewski On Simple Char acterizations of Sheffer psi - p olynomials and R elate d Pr op ositions of the Calculus of Se quenc es ,Bulletin de 6 la S oc. d es Sciences et de Lettres de Lod z, 52 Ser. Rech. 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