A Family of Generalized Stirling Numbers of the First Kind

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1 Apped Mathematc, 4, 5, Pubhed Oe Jue 4 ScRe. A Famy of Geerazed Strg Number of the Frt Kd Beh S. E-Deouy, Nabea A. E-Bedwehy, Abdefattah Mutafa, Fatma M. Abde Meem Mathematc Departmet, Facuty of Scece, Maoura Uverty, Maoura, Egypt Mathematc Departmet, Facuty of Scece, Dametta Uverty, Dametta, Egypt Ema: b_deouy@yahoo.com Receved 4 March 4; reved 4 Apr 4; accepted Apr 4 Copyrght 4 by author ad Scetfc Reearch Pubhg Ic. Th wor ceed uder the Creatve Commo Attrbuto Iteratoa Lcee (CC BY. Abtract A modfed approach va dffereta operator gve to derve a ew famy of geerazed Strg umber of the frt d. Th approach gve u a eteo of the techque gve by E-Deouy [] ad Goud []. Some ew combatora dette ad may reato betwee dfferet type of Strg umber are foud. Furthermore, ome teretg peca cae of the geerazed Strg umber of the frt d are deduced. Ao, a coecto betwee thee umber ad the geerazed harmoc umber derved. Fay, ome appcato coheret tate ad matr repreetato of ome reut obtaed are gve. Keyword Strg Number, Comtet Number, Creato, Ahato, Dffereta Operator, Mape Program. Itroducto Goud [] proved that where (, ad ( d e D e, D e, D, D D, d ( ( ( ( (, are the uua Strg umber ad the ge Strg umber of the frt d, repectvey, defed by,,,, ad, for. ( ( ( ( δ ( > (,, (, δ, ad (, for, (3 > How to cte th paper: E-Deouy, B.S., E-Bedwehy, N.A., Mutafa, A. ad Meem, F.M.A. (4 A Famy of Geerazed Strg Number of the Frt Kd. Apped Mathematc, 5,

2 B. S. E-Deouy et a. ( ( ( ad ( ( where Thee umber atfy the recurrece reato. ( ( (,,,, (4 ( ( (,,,. (5 EL-Deouy [] defed the geerazed Strg umber of the frt d ( ; r,, caed (, umber of the frt d by ( ; r, for < or umber ad ( r -Strg r r ( r r e D e D e D e ( ; r, D, (6 > ad ( ;,, r where ( r r r :,,, a equece of oegatve teger. Equato (6 equvaet to r a equece of rea :,,, ra r ( a ra r a e a e a e a e ( ; r, a, (7 where a ad a are boo creato ad ahato operator, repectvey, ad atfy the commutato rea- to a, a. ; r, atfy the recurrece reato The umber ( ( ; r r, r ( ;,, r (8 wth the otato r r ( r, r,, r ad (,,, The umber ( ; r, have the epct formua where, wth ad. σ. ( ; r, r, (9 σ β, β Moreover E-Deouy [] derved may peca cae ad ome appcato. For the proof ad more deta, ee []. The geerazed fag factora of aocated wth the equece : (,,, of order, where,,, are rea umber, defed by (, (, (,. Comtet [3] [4] ad [5] defed (, the geerazed Strg umber of the frt d, whch are caed Comtet umber, by Thee umber atfy the recurrece reato E-Deouy ad Cac [6] defed (, ; where ( ( (,,. ( ( ( (,,,. (, the geerazed Comtet umber by ( ( ( δ δ, ; δ, ( for, (,; ( ( ad (,,,,, ; >. For more deta o geerazed Strg umber va dffereta operator, ee [7]-[] ad []. 574

3 B. S. E-Deouy et a. The paper orgazed a foow: r I Secto, ug the dffereta operator ( e e ( e e ( e e r r r r r D D D of geerazed Strg umber of the frt d, deoted by ( ;, we defe a ew famy r. A recurrece reato ad a epct formua of thee umber are derved. I Secto 3, ome teretg peca cae are dcued. Moreover ome ew combatora dette ad a coecto betwee ( r, ;, ad the geerazed harmoc umber ( O are gve. I Secto 4, ome appcato coheret tate ad matr repreetato of ome reut obtaed are gve. Secto 5 devoted to the cocuo, whch hade the ma reut derved throughout th wor. Fay, a computer program wrtte ug Mape ad eecuted for cacuatg the geerazed Strg umber of the frt d ad ome peca cae, ee Apped.. Ma Reut Let r : ( r, r,, r be a equece of rea umber ad : (,,, be a equece of oegatve teger. Defto. The geerazed Strg umber ( ; r, are defed by β ( r r r r r r r e D e e D e e D e e ( ; r, D, (3 ( ( ( ( where β, ; r, for β Equato (3 equvaet to Theorem. The umber ( ; r, > ad ( r ;,. ( ( ( ( β r a ( ra ra r a r a ra ra e a e e a e e a e e ; r, a. (4 atfy the recurrece reato ( ; r r, r r ( ;,, r (5 wth the otato r r : ( r r r : (,,, ad,,,. β ( ( r β r r r e ( ; r, D ( e D e e ( ; r, D β ( r r r e D e ( ; r, D β ( r m e D r r ( m;, D r m β ( r m e r r ( m; r, D m β ( r e r r ( ; r, D. Equatg the coeffcet of D o both de yed (5. Theorem. ; r, have the epct formua The umber ( ( r σ σ β, ;, r r, where, r. (6 575

4 B. S. E-Deouy et a. r r r r r e D e e e ( D ri e ( r D, e e e e e e e e e r ( r r e e ( r ( D ( r r I D ( r r e ( r ( r r D, ( ( ( r r r r r r r r ( r r D D D r D D ( r D thu, by terato, we get Settg β σ, r r r r r r (e D e (e D e (e D e ( r e r r D, r. we obta (7 β ( r r r r r r r e D e e D e e D e e r r D. σ β, (8 ( ( ( Comparg (3 ad (8 yed (6. 3. Speca Cae Settg r rad,,, (3, we have the foowg defto. Defto 3. r, ;,, be defed by For ay rea umber r ad oegatve teger, et the umber ( where (,; r, ad ( ( ( D r r, ;, for >. Equato (9 equvaet to ( r r e e e r, ;, D, (9 ( ( a r a Coroary 3. r, ;, atfy the recurrece reato The umber ( ( ra ra e e e r, ;, a. ( (, r ;, ( ( r (, r ;,. ( The proof foow drecty from Equato (5 by ettg r r ad,,,,. Coroary 3. r, ;, have the epct formua The umber ( By ubttutg r ( r, ;, r (. ( σ, r ad,,,, Equato (7, yed 576

5 B. S. E-Deouy et a. the ettg σ we have r σ σ e D e e ( r D,,, r r ( ( r r r e D e e r D, σ, ( ( hece comparg Equato (9 ad (3 we obta Equato (. Furthermore we hade the foowg peca cae. If r, the we have Defto 3. where (, ;, ad ( ( (3 ( (, ;, for >. Coroary 3.3, ;, atfy the recurrece reato The umber ( : The proof foow drecty from Equato ( by ettg r. Coroary 3.4, ;, have the epct formua The umber ( The proof foow drecty from Equato ( by ettg r. If, the we have Defto 3.3 r, ;, are defed by The umber ( where (,; r, δ, ad ( ( e De e, ;, D, (4 (, ;, ( (, ;,. (5 (, ;, (. (6 σ, ( ( r ( r r e De e r, ;, D, (7 r, ;, for >. Coroary 3.5 r, ;, atfy the traguar recurrece reato The umber ( ( r ( r ( r ( r, ;,, ;,, ;,. (8 The proof foow eay from ( by ettg. Coroary 3.6 r, ;, have the foowg epct formua The umber ( ( r r (, ;,. (9 σ, {,} The proof foow from ( by ettg. Ao, ug the recurrece reato (8 we ca fd the foowg epct formua. 577

6 B. S. E-Deouy et a. Theorem 3. The umber ( r, ;, For {,} have the foowg epct epreo For, we get ( r 3r r ( r, ;,. (3, {,}, ( r ( r( r( r ( ( r r ( ( ( r, ;,, ;, ( r ( r( r ( r 3r 3 r, {,} ( ( r 3r 3 r, {,}, ;,, ;,. That the ame recurrece reato (8 for the umber ( r If r ad, the we have Defto 3.4, :, ;, are defed by The umber ( ( where (, δ, ad ( Equato (3 equvaet to, for >. ( D (, ;,. Th compete the proof. e e e, D, (3 ( D ( a a a e e e, a. (3 Coroary 3.7, atfy the traguar recurrece reato The umber ( The proof foow by ettg r Equato (8. Coroary 3.8, have the epct formua The umber ( ( ( ( (,,,. (33 ( (,. (34 σ, {,} The proof foow by ettg r Equato (9. Moreover (, have the foowg epct formua. Coroary 3.9, have the foowg epct epreo The umber ( The proof foow by ettg r (3. r ( 3 5 (,. (35, {,} 3 578

7 B. S. E-Deouy et a. From Equato (9 ad (3 (ao from Equato (34 ad (35 we have the combatora dette 3 ( r r r r (, {, }, {,} From Equato (9 ad (34 we obta that. (36 σ 3 ( (, {, }, {,}. (37 σ ( r r (, ;,,. (38 Remar 3. Operatg wth both de of Equato (3 o the epoeta fucto e, we get β ( ( ( ( r r r r r r r ;,. Therefore, ce a ozero poyoma ca have oy a fte et of zero, we have If, we obta β rm r ( r r m ;,,. (39 Remar 3. From reato (39, by repacg β ( ; r, r r. (4 m m wth, ad reato (8 we cocude that ( ; r, (, ;, where rm r,,,,. (4 m Th gve u a coecto betwee ( ; r, ad (, ;, the geerazed Comtet umber, ee [6]. Settg r r ad,,,, (39, we get (( ( r r, ;,, (4 hece, we have ( r, ;, (, ;, where ( If, the r,,,,, ee [6]. ( ( ( r, ;, r,. (43 Net we dcu the foowg peca cae of (4 ad (43: If r, the ( ( (, ;,, ( ( (, ;,,, hece we have (, ;, (,, the geerazed Comtet umber, where (,,,, ee [6]. If, the we have, (44 579

8 B. S. E-Deouy et a. ( ( ( r r, ;,, ( ( ( r, ;, r,. hece we obta ( r, ;, (,, Comtet umber, where ( [4]. For eampe f 3, r ad (43 we have 6 ( ( (45 r,,,,, ee [3] ad 3, ;, 4 3. (46 Ug Tabe, L.H.S. of (46 (3,;, (3,;, (3,;, (3,3;, (3,4;, (3,5;, (3,6;, R.H.S. of (46 ( 4 3 ( 3 ( 7 ( Th cofrm (46 ad hece (43. Aother eampe f, r ad 3 (43 we have 6 ( ( 3, ;, (47 Ug Tabe 3, L.H.S. of (47 (,;,3 (,;,3 (,;,3 (,3;,3 (,4;,3 (,5;,3 (,6;, R.H.S. of (46 ( ( 3 3 ( Th cofrm (43. If r, the we get ( ( (,, (, (!,, hece we have (, (,, (,,,,, Settg e t, we have ( Ug, ee [], the Equato (49 yed whch a peca cae of Comtet umber, where ee [3] ad [4] ad Tabe. D : dd t dd t td δ, the ubttutg (. t become ( ( ( t t β ( r r r r r r r t δ t t δ t t δ t e ( ; r, δ. (49 t t t t ( ( (, ( ( F δ f F δ f β t t t t ( r r r ( r r ( r ( r δ δ δ ;, δ. (5 Comparg th equato wth Equato (4. [6], we get where ( m r r,,,, m ( r ( ad ( (48 ;,, ;, (5, ;, are the geerazed Comtet umber of the frt 58

9 B. S. E-Deouy et a. d. Furthermore, ug our otato, t eay from Equato (4.4 [6] ad (4 to how that where ( m ( ( r ( S,;,, ;, S,,, (5 ad (, r r,,,, m Net, we fd a coecto betwee ( r, ;, O whch are defed by, ee [3] ad [4], From (4, we have (, ;, r S are the Strg umber of the ecod d. ad the geerazed harmoc umber O ( ( ( ( r ( ( r ( ( r ep og ( ( r ( ( r. ( ( ( ( r ep og ( ( r ( ( r ep ( ( r ( r r ep O r O! ( ( ( ( O O ( ( r (! r ( ( O O ( ( r (! r. Equatg the coeffcet of o both de, we obta ( ( ( ( ( O O! r ( r r ( (, ;,. (53 From ( ad (53, we have the combatora detty ( ( O O ( ( ( σ,! r hece, ettg, we get the detty 4. Some Appcato. ( ( O O ( ( (.! σ, 4.. Coheret State ad Norma Orderg Coheret tate pay a mportat roe quatum mechac epecay optc. The ormay ordered form of the boo operator whch a the creato operator a tad to the eft of the ahato operator a. Ug P ad geerazed the properte of coheret tate we ca defe ad repreet the geerazed poyoma ( r, (54 (55 58

10 B. S. E-Deouy et a. umber P r, a foow. Defto 4. The geerazed poyoma P ( r, ad the geerazed umber P r, For coveece we appy the coveto defed by r, β ( ( r P ;,, (56 r, r, β ( ( P P ; r,. (57 ( r ;, for < or > β. (58 Now we come bac to orma orderg. Ug the properte of coheret tate, ee [7], the coheret tate ma- P tr eemet of the boo trg yed the geerazed poyoma ( Defto 4. ra ra ra ra ra ra ( e e ( e e ( e e z a a a z β * * ( β ( ( r a r z r z z e ( ; r, a z e ( ; r, z z z e (, r, z * * ( ( r z r z z e Pr, ( z e P r,, where β. * z We defe the poyoma (, ad the umber P a For coveece we appy the coveto r, ( ( (59 P,,, (6 P P,,. (6 ( ( ( ( δ ( P( P(, ad, for > ad,. (6 Smary, ug the properte of coheret tate ad (3 we have 4.. Matr Repreetato, ( e e e (, z a a a a z z a z * * z z ( ( e, z zz e, z (63 * * z z z e P( z, e P,. z * I th ubecto we derve a matr repreetato of ome reut obtaed. Let r be ower trage matr, where r.e. r (, the matr whoe etre are the umber ( r, ;,, r ;,. Furthermore et N be a ower trage matr defed by r, 58

11 B. S. E-Deouy et a. ( r r N r e, ;,, M r a dagoa matr whoe etre of the ma dagoa are, ( T r r 4r r r r r r r r.e. Mr dag( e,e,e,,e, Rr,,, (,,,, T D D D D D. Equato (7, may be repreeted a matr form a for eampe f 3 the t vere gve by r r r r ( r r r r ad e,,,, r, R ND M D, (64 r r r r r e D e De r r re e D 4r 4r 4r 3r e 4e r e D 3 6r 6r 6r 6r 3 5r e 3r e 9re e D 3 Settg r (64, we get hece ( r e D r e r D, 4r e 3r 4r D 6r 3 3 e 5r 3r 9r D r r r r r (65 D N R M R. (66 R ND M D (67, e D e D e De e e e D D , 3e 4e e D e 3 4 D e 3e 9e e D e D 3 For 3, we have D N R M R ( D. ( D e e e e D e e D e 4 4 D 4e e 4 e D 3e 9e e 3 9 e 3 5. Cocuo 3 (68 (69 I th artce we vetgated a ew famy of geerazed Strg umber of the frt d. Recurrece reato ad a epct formua of thee umber are derved. Moreover ome teretg peca cae ad ew 583

12 B. S. E-Deouy et a. combatora dette are obtaed. A coecto betwee th famy ad the geerazed harmoc umber gve. Fay, ome appcato coheret tate ad matr repreetato of ome reut are obtaed. Referece [] E-Deouy, B.S. ( Geerazed Strg Number of the Frt Kd: Modfed Approach. Joura of Pure ad Mathematc: Advace ad Appcato, 5, [] Goud, H.W. (964 The Operator ( a ad Strg Number of the Frt Kd. The Amerca Mathematca Mothy, 7, [3] Comtet, L. (97 Nombre de trg géérau et focto ymétrque. Compte Redu de Académe de Scece (Sere A, 75, [4] Comtet, L. (974 Advaced Combatorc: The Art of Fte ad Ifte Epato. D. Rede Pubhg Compay, Dordrecht, Hoad. [5] Comtet, L. (973 Ue formue epcte pour e puace ucceve de operateur dervato de Le. Compte Redu de Académe de Scece (Sere A, 76, [6] E-Deouy, B.S. ad Cać, N.P. ( Geerazed Hgher Order Strg Number. Mathematca ad Computer Modeg, 54, [7] Baa, P. (5 Combatorc of Boo Norma Orderg ad Some Appcato. PhD The, Uverty of Par, Par. [8] Baa, P., Peo, K.A. ad Soomo, A.I. (3 The Geera Boo Norma Orderg Probem. Phyc Letter A, 39, [9] Cac, N.P. (98 O Some Combatora Idette. Appcabe Aay ad Dcrete Mathematc, , [] Cartz, L. (93 O Array of Number. Amerca Joura of Mathematc, 54, [] E-Deouy, B.S., Cac, N.P. ad Maour, T. ( Modfed Approach to Geerazed Strg Number va Dffereta Operator. Apped Mathematc Letter, 3, [] Vov, O.V. ad Srvatava, H.M. (994 New Approache to Certa Idette Ivovg Dffereta Operator. Joura of Mathematca Aay ad Appcato, 86, -. [3] Macdoad, I.G. (979 Symmetrc Fucto ad Ha Poyoma. Caredo (Oford Uverty Pre, Oford, Lodo ad New Yor. [4] Cho, J. ad Srvatava, H.M. ( Some Summato Formua Ivovg Harmoc Number ad Geerazed Harmoc Number. Mathematca ad Computer Modeg, 54,

13 B. S. E-Deouy et a. Apped Tabe of (, ; r, cacuated ug Mape, for ome vaue of,, r ad : Tabe., 4, r Tabe., 4, r Tabe 3., 4, r, ad Notce that the at coum a tabe ut the um of the etre of the correpodg row. 585