De la Vallée Poussin Summability, the Combinatorial Sum 2n 1

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1 J o u r a l of Mathematics ad Applicatios JMA No 40, pp 5-20 (2017 De la Vallée Poussi Summability, the Combiatorial Sum 1 ( 2 ad the de la Vallée Poussi Meas Expasio Ziad S. Ali Abstract: I this paper we apply the de la Vallée Poussi sum to a combiatorial Chebyshev sum by Ziad S. Ali i [1]. Oe outcome of this cosideratio is the mai lemma provig the followig combiatorial idetity: with Re(z stadig for the real part of z we have 1 2 Re ( 2F 1 (1, 1/2 + ; 1 + ; 4 4 2F 1 (1, 1/2 + ; 1 + ; 4. Our mai lemma will idicate i its proof that the hypergeometric factors 2F 1 (1, 1/2 + ; 1 + ; 4, ad 2 F 1 (1, 1/2 + ; 1 + ; 4 are complex, each havig a real ad imagiary part. As we apply the de la Vallée Poussi sum to the combiatorial Chebyshev sum geerated i the Key lemma by Ziad S. Ali i [1], we see i the proof of the mai lemma the extreme importace of the use of the mai properties of the gamma fuctio. This represets a secod importat cosideratio. A third ew outcome are two iterestig idetities of the hypergeometric type with their ew Meijer G fuctio aalogues. A fourth outcome is that by the use of the Cauchy itegral formula for the derivatives we are able to give a differet meaig to the sum: 1 2. COPYRIGHT c by Publishig House of Rzeszów Uiversity of Techology P.O. Box 85, Rzeszów, Polad

2 6 Z.S. Ali A fifth outcome is that by the use of the Gauss-Kummer formula we are able to mae better sese of the expressios 4 2F 1 (1, 1/2 + ; 1 + ; 4, ad 2F 1 (1, 1/2 + ; 1 + ; 4 by maig use of the series defiitio of the hypergeometric fuctio. As we cotiue we otice a ew close relatio of the Key lemma, ad the de la Vallée Poussi meas. With this close relatio we were able to tal about the de la Vallée Poussi summability of the two ifiite series 0 cos θ, ad 0 ( 1 cos θ. Furthermore the applicatio of the de la Vallée Poussi sum to the Key lemma has created two ew expasios represetig the followig fuctios: ad 2 ( 1 (1 + x ( (1 + x, where x cos θ, (2x ( 1 ( (1 x (1 x, where x cos θ (2x 1 i terms of the de la Vallée Poussi meas of the two ifiite series cos θ, ad 0 ( 1 cos θ. 0 AMS Subject Classificatio: 47H09, 47H14, 54E5. Keywords ad Phrases: Complete metric space; Hyperbolic space; Ifiite product; Noexpasive mappig; Radom wea ergodic property. 1. Itroductio The Gauss Hypergeometric fuctio is give by: 2F 1 (a, b; c; z 1 + ab a(a + 1b(b + 1 z + 1 c 1 2 c(c + 1 z (a (b (c z!, where the above series coverges for z < 1 ad (a is is the Pochhammer symbol defied by: (a 0 0, ad (a a(a + 1(a (a + 1.

3 De la Vallée Poussi summability 7 We further ote that: (a Γ(a +, Γ(a where the symbol Γ refers to the gamma fuctio. Now we lie to brig up two defiitios related to de la Vallée Poussi; oe is related to the de la Vallée Poussi meas, ad the other is related to the de la Vallée Poussi sum. We have: the de la Vallée Poussi meas of the ifiite series are defied by (see []: V (, a j0 0 a (! 2 ( j!( + j! a j. Let T i be the Chebyshev polyomials. The Charles Jea de la Vallée-Poussi defies (see [2] the de la Vallée Poussi sum SV as follows: where SV S + S S 1, S 1 2 c 0(f + c j (ft j. We shall refer to the S as the Chebyshev sum or the Chebyshev expasio of f. The very importat properties of the gamma fuctios that were used i this wor, ad helped immesely i the proof of the mai lemma are: j1 Γ(z + 1 zγ(z, ad Γ(zΓ(1 z π si πz. Let f by holomorphic i a ope subset U of the complex plae C, ad let z z 0 r, be cotaied completely i U. Now let a be ay poit iterior to z z 0 r. The the Cauchy itegral formula for the derivative says that the - th derivative of f at a is give by: f (a! f(z dz. 2πi z z 0 r (z a +1 The Meijer G fuctio is defied by G m, a1,..., a p p,q ; z 1 m j1 Γ(b j s j1 Γ(1 a j + s b 1,..., b q 2πi q L jm+1 Γ(1 b j + s p j+1 Γ(a j s zs ds.

4 8 Z.S. Ali The L i the itegral represets the path of itegratio. We may choose L to ru from i to +i such that all poles of Γ(b j s, j 1, 2,..., m, are o the right of the path, while all poles of Γ(1 a + s, 1, 2,...,, are o the left. The Meijer G fuctio, ad the Hypergeometric fuctio p F q are related by: q a1,..., a p 1 pf q ; z Γ(b 1 a1,..., 1 a p b 1,..., b p q 1 Γ(a G1,p p,q 1 ; z. 0, 1 b 1,... 1 b q 2. The close relatio betwee the Key lemma ad the de la Vallée Poussi meas I [1] we have the followig Key lemma: Lemma 2.1. For 1 r, ad θ real we have: (cos (i rθ + ( 1 r (1 + cos θ. r (ii r1 ( 1 r cos rθ (1 cos θ. r 2 r1 Before we move to the mai lemma ad its proof we state two theorems which are direct cosequeces of the Key lemma. We lie to idicate before we state them that they are related to the de la Vallée Poussi meas of the the followig two ifiite series: cos θ, ad ( 1 cos θ. 0 Clearly from the Key lemma we have : 0 cos rθ (1 + cos θ, r 2 r0 ( 1 r cos rθ (1 cos θ. r 2 r0 Accordigly with V (, cos θ beig the de la Vallée Poussi meas of cos θ, 0

5 De la Vallée Poussi summability 9 ad V (, ( 1 cos θ beig the de la Vallée Poussi meas of We have: ( 1 cos θ. 0 V (, cos θ 1 (1 + cos θ ( + 1 2, ad V (, ( 1 cos θ 1 (1 cos θ ( Now it ca be show that for each real θ 2π, where is a iteger the followig limit: 2 1 (1 + cos θ lim ( 0, ad for each real θ π,where is a odd iteger the followig limit 2 1 (1 cos θ lim ( 0. Oe way of showig the above limits is the use of the Stirlig s formula for the gamma fuctio which says: Γ(x e x (2πx 1/2+x as x. Accordigly we have the followig two theorems: Theorem 2.2. For each real θ 2π, where is a iteger the ifiite series cos θ 0 is summable i the sese of de la Vallée Poussi to 1 2. Theorem 2.. For each real θ π, where is a odd iteger the ifiite series ( 1 cos θ 0 is summable i the sese of de la Vallée Poussi to 1 2.

6 10 Z.S. Ali. The mai lemma There are two versios of the mai lemma. We give ow oe versio, ad we will provide the other oe at the ed of the curret sectio. Lemma.1. Let 2 F 1 (a, b; c; z be the Gauss Hypergeometric fuctio, the we have: 1 2 2F 1 (1, 1/2 + ; 1 + ; 4 4 2F 1 (1, 1/2 + ; 1 + ; 4. Before we prove the above lemma we lie to idicate that by the Gauss-Kummer formula it is easily see that the hypergeometric fuctios 2 F 1 (1, 1/2 + ; 1 + ; 4, ad 2 F 1 (1, 1/2 + ; 1 + ; 4 are both discrete complex valued fuctios, ad they are iterestig as for exmple the left had side of the idetity give above is a atural umber, which leads us to say that the imagiary part of the right had side of the idetity above is actually zero. Aother iterestig view is that the secod term i the above lemma is obtaied by replacig i the first term by. The further iterestig ideas ivolved are comig as we cotiue to prove the above lemma, ad cotiue further. Proof of the case 1 of Lemma.1. By iductio o. We ote first that for 1 we have: F 1 (1, /2; 2; F 1 (1, 5/2; ; 4. Now oe way to evaluate the above expressio is by usig a already ow techique, where the umbers 1, /2, 2, 4 iside the hypergeometric fuctio 2 F 1 for example i this case are fed ito their proper slot of a computer program or a plug i algorithm which evaluates the hypergeometric fuctios of the type give above to get the complex umber represetig 2 F 1 (1, /2; 2; 4. Similarly the umbers 1, 5/2,, 4 are fed ito each slot i the computer program to get the complex umber represetig 2F 1 (1, 5/2; ; 4. For example we see by usig this method we have: 2 ( 1 2 i 6 6 ( 1 2 i 18 ad the idetity give i the lemma is true for 1. Oe way to prove the case for 1 without the use of a computer is by the use of the Gauss-Kummer formula, which states: for (a b ot a iteger, ad z ot i the uit iterval (0, 1 we have 2F 1 (a, b; c; z + Γ(b aγ(c Γ(bΓ(c a ( z a 2F 1 (a, a c + 1; a b + 1; 1/z Γ(a bγ(c Γ(aΓ(c b ( z b 2F 1 (b, b c + 1; a + b + 1; 1/z.,

7 De la Vallée Poussi summability 11 We will ow do basic calculatios for the proof of Corollary.2, ad Corollary. comig up. Accordigly we have: 2F 1 (1, /2; 2; 4 Γ(1/2Γ(2 Γ(/2Γ(1 ( 4 1 2F 1 (1, 0; 1/2; 1/4 + Γ( 1/2Γ(2 Γ(1Γ(1/2 ( 4 /2 2F 1 (/2, 1/2; /2; 1/4. Now sice we have: Γ(1/2 π, Γ(/2 π 2, Γ( 1/2 2 π Note that Now sice 2F 1 (1, /2; 2; 4 Γ(1/2Γ(2 Γ(/2Γ(1 ( 4 1 2F 1 (1, 0; 1/2; 1/ (0 (1 (1/2 4! F 1 (/2, 1/2; /2; 1/4 (1 (1/4 2 1/2, ad ( 4 /2 i 8, we have: Γ( 1/2Γ(2 Γ(1Γ(1/2 ( 4 /2 2F 1 (/2, 1/2; /2; 1/4 i 6. We lie to ote ow that by usig the biomial expasio formula we have for z < 1: ( z + 1 1/2 Similarly we have: (x + a γ 0 γ x a γ, ( 1 ( 1 z 2 2 F 1 (/2, 1/2; /2; 1/4. 0 2F 1 (1, 5/2; ; 4 Γ(/2Γ( Γ(5/2Γ(2 ( 4 1 2F 1 (1, 1; 1/2; 1/4 + Γ( /2Γ( Γ(1Γ(1/2 ( 4 5/2 2F 1 (5/2, 1/2; 5/2; 1/4,

8 12 Z.S. Ali which simplifies to: Note that i F 1 (1, 1; 1/2; 1/4 1 2 i (1 ( 1 ( 1/2 4! 2. For 2 the term ( is preset i each product of ( 1 : accordigly we are addig the first two terms oly; furthermore 2F 1 (5/2, 1/2; 5/2; 1/4 2. This is the case where a c 5/2, ad i this case the sum is /4 2 This completes the iductio proof for the case 1 without the use of a computer. We ca clearly see from above that 2 F 1 (1, /2; 2; 4, ad 2 F 1 (1, 5/2; ; 4 are complex umbers, while 2 F 1 (1, 0; 1/2; 1/4, 2 F 1 (/2, 1/2; /2; 1/4, 2 F 1 (1, 1; 1/2; 1/4, ad 2 F 1 (5/2, 1/2; 5/2; 1/4 are real umbers. From above we have the followig Corollaries resultig from the iductio proof whe 1, ad i relatio to the real part, ad the imagiary part of 2 F 1 (1, /2; 2; 4, ad 2 F 1 (1, 5/2; ; 4.. Corollary.2. Let i be the imagiary uit, let Re(z, ad Im(z respectively deote the real part, ad the imagiary part of z. We have: ad Re 2 F 1 (1, /2; 2; 4 Γ(1/2Γ(2 Γ(/2Γ(1 ( 4 1 2F 1 (1, 0; 1/2; 1/4 1 2, i Im 2 F 1 (1, /2; 2; 4 Γ( 1/2Γ(2 Γ(1Γ(1/2 ( 4 /2 2F 1 (/2, 1/2; /2; 1/4 i 6.

9 De la Vallée Poussi summability 1 Corollary.. Let i be the imagiary uit, let Re(z, ad Im(z respectively deote the real part, ad the imagiary part of z. We have: ad Re 2 F 1 (1, 5/2; ; 4 Γ(/2Γ( Γ(5/2Γ(2 ( 4 1 2F 1 (1, 1; 1/2; 1/4 1 2, i Im 2 F 1 (1, 5/2; ; 4 Γ( /2Γ( Γ(1Γ(1/2 ( 4 5/2 2F 1 (5/2, 1/2; 5/2; 1/4 i Accordigly we have the followig geeral lemmas: Lemma.4. Let i be the imagiary uit, let Re(z, ad Im(z respectively deote the real part, ad the imagiary part of z. We have: Re 2 F 1 (1, 1/2 + ; 1 + ; 4 i Im 2 F 1 (1, 1/2 + ; 1 + ; 4 Γ( 1/2Γ( + 1 ( 4 1 2F 1 (1, 1 ; /2 ; 1/4 Γ( + 1/2Γ( Γ(1/2 Γ( + 1 ( 4 1/2 Γ(1Γ(1/2 2 F 1 (1/2 +, 1/2; 1/2 + ; 1/4. The proof of the above lemma follows by usig the Gauss-Kummer formula, ad by the presece of ( 4 1/2 i the secod equatio. Lemma.5. Let i be the imagiary uit, let Re(z, ad Im(z respectively deote the real part, ad the imagiary part of z. We have: Re 2 F 1 (1, 1/2+; 1+; 4 i Im 2 F 1 (1, 1/2 + ; 1 + ; 4 Γ( 1/2Γ( + 1 ( 4 1 2F 1 (1, 1 ; /2 ; 1/4 Γ( + 1/2Γ( Γ(1/2 Γ( + 1 ( 4 1/2 Γ(1Γ(1/2 2 F 1 (1/2 +, 1/2; 1/2 + ; 1/4. The proof of the above lemma follows by usig the Gauss-Kummer formula, ad by the presece of ( 4 1/2 i the secod equatio. Now we have the followig lemma showig that the imagiary part of the right had side of the mai lemma above equals to zero, i.e.:

10 14 Z.S. Ali Lemma.6. Let i be the imagiary uit, ad let Im(z deote the imagiary part of z. We have: Im ( 2F 1 (1, 1/2 + ; 1 + ; 4 4 2F 1 (1, 1/2 + ; 1 + ; 4 0. Proof of Lemma.6. Γ(1/2 Γ( + 1 ( 4 1/2 2F 1 (1/2 +, 1/2; 1/2 + ; 1/4 Γ(1Γ(1/2 4 Γ(1/2 Γ( + 1 ( 4 1/2 2F 1 (1/2 +, 1/2; 1/2 + ; 1/4 0, Γ(1Γ(1/2 clearly 2F 1 (1/2+, 1/2; 1/2+; 1/4 2 F 1 (1/2+, 1/2; 1/2+; 1/ /4 Now by otig that 2 Γ( + 1/2 πγ( + 1 ad 4 24 Γ( + 1/2, πγ( + 1 what we do ow is to substitute the just above idetities for (, ad 4, i the above equatio whose right had side is zero; the we use the complemet formula of the gamma fuctio which is: Γ(zΓ(1 z π si πz to hadle both of the followig products: Γ(1/2 + Γ(1/2 ad Γ(1/2 + Γ(1/2. Fially by writig ( 4 1/2 j (2 2 e iπ 1/2 j, with j 1, 2. We ca easily show that the stated lemma above is correct, ad that the imagiary part of the right had side of the mai lemma is idetically equal to zero. This completes the proof of the above lemma.

11 De la Vallée Poussi summability 15 Proof of the iductio step. Sice the imagiary part of the right had side of the mai lemma give above equals zero. We have: 4 2F 1 (1, 1/2 + ; 1 + ; 4 2F 1 (1, 1/2 + ; 1 + ; 4 Γ( 1/2Γ( + 1 ( 4 1 2F 1 (1, 1 ; /2 ; 1/4 Γ( + 1/2Γ( 4 Γ( 1/2Γ( + 1 ( 4 1 2F 1 (1, 1 ; /2 ; 1/4. Γ( + 1/2Γ( Now assume that the idetity we are to prove is true for, we eed to show that it is true for + 1; i.e. we eed to show that: F 1 (1, /2+; 2+; equivaletly we eed to show that: F 1 (1, 5/2+; +; Γ( + 1/2Γ( Γ( + /2Γ( + 1 ( 4 1 2F 1 (1, ; 1/2 ; 1/ Γ( + /2Γ( Γ( + 5/2Γ( + 2 ( 4 1 2F 1 (1, 1 ; 1/2 ; 1/4 equivaletly agai we eed to show that: + 2 Γ( + 1/2Γ( Γ( + /2Γ( + 1 ( 4 1 2F 1 (1, ; 1/2 ; 1/ Γ( + /2Γ( Γ( + 5/2Γ( + 2 ( 4 1 2F 1 (1, 1 ; 1/2 ; 1/4 Γ( 1/2Γ( + 1 ( 4 1 2F 1 (1, 1 ; /2 ; 1/4 ( 4 Γ( + 1/2Γ( Γ( 1/2Γ( + 1 ( 4 1 2F 1 (1, 1 ; /2 ; 1/4 Γ( + 1/2Γ( The above iductio step ca be show by otig the followig two ew hypergeometric idetities: 2F 1 (1, ; 1/2 ; 1/4 4( F 1 (1, 1 ; /2 ; 1/4 + 1,

12 16 Z.S. Ali F 1 (1, 1 ; /2 ; 1/ F 1 (1, 1 ; (1/2 ; 1/ The proof of the first hypergeometric idetity above is simple, ad the proof of the secod hypergeometric idetity above is oe where oe really appreciates the simple law of cacellatio. Based from above, we provide the secod versio of the mai lemma: Lemma.7. Let Re(z be the real part of z, the we have: 1 2 Re ( 2F 1 (1, 1/2 + ; 1 + ; 4 4 2F 1 (1, 1/2 + ; 1 + ; 4. A iterestig corollary is the followig: Corollary.8. Let f( 2 Γ( + 1/2Γ(1/2 ( 4 1/2 π, the f( f(. 4. Presetatio of the two ew hypergeometric idetities above i terms of the Meijer G fuctio Sice the followig idetities relatig the hypergeometric fuctio 2 F 1, ad the Meijer G 1,2 2,2 fuctio, hold: Γ(1/2 2F 1 (1, ; 1/2 ; 1/4 G 1,2 0 (1 + 2,2 Γ( 0 (1/2 + ; 1/4, 2F 1 (1, 1 ; /2 ; 1/4 Γ(/2 Γ(1 G 1,2 0 2,2 0 ( 1/2 + ; 1/4, 2F 1 (1, 1 ; /2 ; 1/4 Γ(/2 0 Γ(1 G1,2 2,2 0 ( 1/2 + ; 1/4,

13 De la Vallée Poussi summability 17 2F 1 (1, 1 ; 1/2 ; 1/4 Γ( 1/2 Γ( 1 G 1,2 0 ( + 2 2,2 0 (/2 + ; 1/4 the we have the followig two ew idetities relatig the Meijer fuctio G 1,2 Γ(1/2 G 1,2 2,2 Γ( 0 (1 + 0 (1/2 + ; 1/4 G 1,2 0 2,2 0 ( 1/2 + ; 1/4 + 1, Γ(/2 0 Γ(1 G1,2 2,2 0 ( 1/2 + ; 1/4 G 1,2 2,2 ( ( 0 ( + 20 (/2 + ; 1/4 ( A differet meaig of 1 Γ(/2 4(1/2 Γ(1 2,2 : 4 1 Γ( 1/2 Γ( ( 2 Let r > 0 be give: m 1 (1 + z m r 2πi z r z r+1 dz. Accordigly: 2 1 (1 + z 2 2πi z r z +1 dz. Hece 1 1 (1 + z 2 dz 2πi z r (z F 1 (1, 1/2 + ; 1 + ; 4 4 2F 1 (1, 1/2 + ; 1 + ; 4. Accordigly by the Cauchy itegral formula for derivatives we have the followig ew, ad differet meaig of 1 ( 2 : d ( (1 + z 2! dz. z0 The above formula clearly states the differet meaig of the sum 1 ( 2. We remar that d ( (1 + z 2 dz z0 stads for the th derivative of of (1 + z 2 evaluated at z 0.,

14 18 Z.S. Ali 6. The de la Vallée Poussi sum, ad the de la Vallée Poussi meas expasio I this sectio, we apply the de la Vallée Poussi sum SV to the Key lemma parts (i, ad (ii to obtai a implicit, ad a explicit de la Vallée Poussi meas expasios DV P i, ad DV P ii of two particular fuctios, which will be defied i the statemets of the theorems. We remar that from the Key lemma we ca easily see: S ( j1 2 cos jθ. j Accordigly, we have the followig two theorems: Theorem 6.1. The fuctio 2 ( 1 (1 + x ( (1 + x, where x cos θ (2x + 1 has a implicit de la Vallée Poussi meas expasio DV P i of the form T j (x j ad a explicit de la Vallée Poussi expasio DV P i of the form where V 1 are the de la Vallée Poussi meas of cos θ. 1 j1 1 2 V 1, Theorem 6.2. The fuctio 2 ( 1 ( (1 x (1 x, where x cos θ (2x 1 has a implicit de la Vallée Poussi meas expasio DV P ii of the form ( 1 j T j (x j j1

15 De la Vallée Poussi summability 19 ad a explicit de la Vallée Poussi expasio DV P ii of the form where V 2 are the de la Vallée Poussi meas of 1 ( 1 cos θ. 1 2 V 2, 7. Cocludig remar I this paper, we were able to show that 1 2 Re ( 2F 1 (1, 1/2 + ; 1 + ; 4 4 2F 1 (1, 1/2 + ; 1 + ; 4. Moreover, we were able to create two ew hypergeometric idetities to prove the iductio step. After, it was iterestig to fid their Meijer G fuctio aalogue. Further, by the use of the Key lemma ad the defiitio of the de la Vallée Poussi meas, we were able to fid two ew expasios represetig the followig fuctios: ad 2 ( 1 (1 + x ( (1 + x, where x cos θ (2x ( 1 ( (1 x (1 x, where x cos θ. (2x 1 The geeral form of the expasios ca be put ito a more familiar form as: A K1 A (K, V (K,, where A 0 1 2K + 2, ad K + 1 K1 A (K, 1 2K + 2. K + 1 We lie to add that the strog coectio of the Key lemma, ad the de la Vallée Poussi meas has give us two theorems about the de la Vallée Poussi summability of the two ifiite series cos θ θ 2π iteger, ad 0

16 20 Z.S. Ali ( 1 cos θ θ π odd iteger. 0 Moreover we ote for example that ( 1 0 is also de la Vallée Poussi summable to 1 2 just lie it s Cesàro sum, ad it s Abel sum. We remar fialy that: K1 Refereces ( 2K + 2 Re 2F 1 (1, 1/2 + ; 1 + ; 4 K F 1 (1, 1/2 + ; 1 + ; 4. [1] Ziad S. Ali, O a multipilier of the progressive meas ad covex maps of the uit disc, Acta Mathematica Academiae Paedagogicae Nyiregyhaziesis 27 ( [2] C. Jea de la Vallée-Poussi, Charles Jea de la Vallée-Poussi, WIKIPEDIA The free Ecyclopedia. Jea de la Vallée-Poussi. [] G. Pólya, I.J. Schoeberg, Remars o de la Vallée Poussi meas ad covex coformal maps of the circle, Pacific J. Math. 8 ( DOI: /rf Ziad S. Ali alioppp@yahoo.com America College of Switzerlad Leysi 1854 SWIZERLAND Received Accepted