Every positive real number x has a g-adic representation


Firstly we consider only the case x[0,1[ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaacUfacaaIWaGaaiilaiaaigdacaGGBbaaaa@3BCD@ . For such a number x we use the Gaussian function [ X ] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacaWGybaacaGLBbGaayzxaaaaaa@3838@

 i

[x]=max{n|nx} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaadIhacaGGDbGaeyypa0JaciyBaiaacggacaGG4bGaai4Eaiaad6gacqGHiiIZcqWIKeIOcaGG8bGaamOBaiabgsMiJkaadIhacaGG9bGaeyicI4SaeSijHikaaa@4A10@ . Note that

0x[x]<1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgsMiJkaadIhacqGHsislcaGGBbGaamiEaiaac2facqGH8aapcaaIXaaaaa@3EBE@ .
to define recursively the sequence ( r n (x)) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadkhadaWgaaWcbaGaamOBaaqabaGccaGGOaGaamiEaiaacMcacaGGPaaaaa@3B38@ by

r 1 (x)xg r n+1 (x)( r n (x)[ r n (x)])g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBaaaleaacaaIXaaabeaakiaacIcacaWG4bGaaiykaiabg2da9iaadIhacqGHflY1caWGNbGaey4jIKTaamOCamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccaGGOaGaamiEaiaacMcacqGH9aqpcaGGOaGaamOCamaaBaaaleaacaWGUbaabeaakiaacIcacaWG4bGaaiykaiabgkHiTiaacUfacaWGYbWaaSbaaSqaaiaad6gaaeqaaOGaaiikaiaadIhacaGGPaGaaiyxaiaacMcacqGHflY1caWGNbaaaa@57CF@

and subsequently set  a n (x)[ r n (x)]. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakiaacIcacaWG4bGaaiykaiabg2da9iaacUfacaWGYbWaaSbaaSqaaiaad6gaaeqaaOGaaiikaiaadIhacaGGPaGaaiyxaaaa@410A@ It is easily seen that 0 r n (x)<g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgsMiJkaadkhadaWgaaWcbaGaamOBaaqabaGccaGGOaGaamiEaiaacMcacqGH8aapcaWGNbaaaa@3E3E@ , so that 0[ r n (x)]<g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgsMiJkaacUfacaWGYbWaaSbaaSqaaiaad6gaaeqaaOGaaiikaiaadIhacaGGPaGaaiyxaiabgYda8iaadEgaaaa@3FFE@ is valid as well. Therefore we have  a n (x) D g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakiaacIcacaWG4bGaaiykaiabgIGiolaadseadaWgaaWcbaGaam4zaaqabaaaaa@3D33@ .
 

When proving the following please note that the number xg a 1 (x)= r 1 (x)[ r 1 (x)] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgwSixlaadEgacqGHsislcaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaaiikaiaadIhacaGGPaGaeyypa0JaamOCamaaBaaaleaacaaIXaaabeaakiaacIcacaWG4bGaaiykaiabgkHiTiaacUfacaWGYbWaaSbaaSqaaiaaigdaaeqaaOGaaiikaiaadIhacaGGPaGaaiyxaaaa@4AE5@   is always a member of [0,1[ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaaicdacaGGSaGaaGymaiaacUfaaaa@394C@ .

  1. r n (xg a 1 (x))= r n+1 (x)  for all  n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBaaaleaacaWGUbaabeaakiaacIcacaWG4bGaeyyXICTaam4zaiabgkHiTiaadggadaWgaaWcbaGaaGymaaqabaGccaGGOaGaamiEaiaacMcacaGGPaGaeyypa0JaamOCamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccaGGOaGaamiEaiaacMcacaqGMbGaaei=aiaabkhacaqGGaGaaeyyaiaabYgacaqGSbGaaeyzaiaad6gacqGHiiIZcqWIvesPdaahaaWcbeqaaiabgEHiQaaaaaa@55F1@

    Proof  by induction:

    ►   r 1 (xg a 1 (x))=(xg[ r 1 (x)])g= r 2 (x) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBaaaleaacaaIXaaabeaakiaacIcacaWG4bGaeyyXICTaam4zaiabgkHiTiaadggadaWgaaWcbaGaaGymaaqabaGccaGGOaGaamiEaiaacMcacaGGPaGaeyypa0JaaiikaiaadIhacqGHflY1caWGNbGaeyOeI0Iaai4waiaadkhadaWgaaWcbaGaaGymaaqabaGccaGGOaGaamiEaiaacMcacaGGDbGaaiykaiabgwSixlaadEgacqGH9aqpcaWGYbWaaSbaaSqaaiaaikdaaeqaaOGaaiikaiaadIhacaGGPaaaaa@57EF@

    ▶   r n+1 (xg a 1 (x))  =( r n (xg a 1 (x))[ r n (xg a 1 (x))])g =( r n+1 (x)[ r n+1 (x)])g = r n+2 (x) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@812E@
     

  2. a n (xg a 1 (x))= a n+1 (x)  for all  n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakiaacIcacaWG4bGaeyyXICTaam4zaiabgkHiTiaadggadaWgaaWcbaGaaGymaaqabaGccaGGOaGaamiEaiaacMcacaGGPaGaeyypa0JaamyyamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccaGGOaGaamiEaiaacMcacaqGMbGaaei=aiaabkhacaqGGaGaaeyyaiaabYgacaqGSbGaaeyzaiaad6gacqGHiiIZcqWIvesPdaahaaWcbeqaaiabgEHiQaaaaaa@55CF@

    The proof  is immediate from 1.
     

  3. For all  n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaaaa@3A68@ the inequality  |x i=1 n a i (x) g i | 1 g n   holds for every  x[0,1[ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsisldaaeWbqaamaalaaabaGaamyyamaaBaaaleaacaWGPbaabeaakiaacIcacaWG4bGaaiykaaqaaiaadEgadaahaaWcbeqaaiaadMgaaaaaaaqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aOGaaiiFaiabgsMiJoaalaaabaGaaGymaaqaaiaadEgadaahaaWcbeqaaiaad6gaaaaaaOGaaeOzaiaabYpacaqGYbGaaeiiaiaabQgacaqGLbGaaeizaiaabwgacaqGZbGaamiEaiabgIGiolaacUfacaaIWaGaaiilaiaaigdacaGGBbaaaa@5941@

    Proof  by induction:

    ►   |x a 1 (x) g |=|x [xg] g |= 1 g |xg[xg]| 1 g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadIhacqGHsisldaWcaaqaaiaadggadaWgaaWcbaGaaGymaaqabaGccaGGOaGaamiEaiaacMcaaeaacaWGNbaaaiaacYhacqGH9aqpcaGG8bGaamiEaiabgkHiTmaalaaabaGaai4waiaadIhacqGHflY1caWGNbGaaiyxaaqaaiaadEgaaaGaaiiFaiabg2da9maalaaabaGaaGymaaqaaiaadEgaaaGaaiiFaiaadIhacqGHflY1caWGNbGaeyOeI0Iaai4waiaadIhacqGHflY1caWGNbGaaiyxaiaacYhacqGHKjYOdaWcaaqaaiaaigdaaeaacaWGNbaaaaaa@5D97@

    ►   For the inductive step we employ the induction hypothesis with the number xg a 1 (x) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgwSixlaadEgacqGHsislcaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaaiikaiaadIhacaGGPaaaaa@3EB6@ !

    |x i=1 n+1 a i (x) g i |  = 1 g |xg a 1 (x) i=1 n a i+1 (x) g i | = 1 g |xg a 1 (x) i=1 n a i (xg a 1 (x)) g i | 1 g 1 g n = 1 g n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@95FC@
     

3. proves the equality  x= i=1 a i (x) g i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2da9maaqahabaWaaSaaaeaacaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadIhacaGGPaaabaGaam4zamaaCaaaleqabaGaamyAaaaaaaaabaGaamyAaiabg2da9iaaigdaaeaacqGHEisPa0GaeyyeIuoaaaa@443A@ . So we only need to overcome the initial restriction for x.
Now let x0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgwMiZkaaicdaaaa@38E6@ by any positive real. With  nmin{k|x< g k } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg2da9iGac2gacaGGPbGaaiOBaiaacUhacaWGRbGaeyicI4SaeSyfHuQaaiiFaiaadIhacqGH8aapcaWGNbWaaWbaaSqabeaacaWGRbaaaOGaaiyFaaaa@4528@   (note: ( g n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadEgadaahaaWcbeqaaiaad6gaaaGccaGGPaaaaa@38D8@ is unbounded) we have x g n [0,1[ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWG4baabaGaam4zamaaCaaaleqabaGaamOBaaaaaaGccqGHiiIZcaGGBbGaaGimaiaacYcacaaIXaGaai4waaaa@3DF3@ and from that we can set

x i a i ( x g n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGPbaabeaakiabg2da9iaadggadaWgaaWcbaGaamyAaaqabaGccaGGOaWaaSaaaeaacaWG4baabaGaam4zamaaCaaaleqabaGaamOBaaaaaaGccaGGPaaaaa@3F16@   for i>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabg6da+iaaicdaaaa@3899@   and in addition  x 0 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIWaaabeaakiabg2da9iaaicdaaaa@3996@ ,

and get:  x g n = i=0 x i g i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWG4baabaGaam4zamaaCaaaleqabaGaamOBaaaaaaGccqGH9aqpdaaeWbqaamaalaaabaGaamiEamaaBaaaleaacaWGPbaabeaaaOqaaiaadEgadaahaaWcbeqaaiaadMgaaaaaaaqaaiaadMgacqGH9aqpcaaIWaaabaGaeyOhIukaniabggHiLdaaaa@4420@   and finally have:

x= i=0 x i g in = i=n x i+n g i = i=n 0 x i+n g i + i=1 x i+n g i = x 0 g n ++ x n g 0 + i=1 x i+n g i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@86E7@