Теореме об ожерелье

[aspair]

Помогите, пожалуйста, доказать теорему произведения чисел ожерелья.

Само число ожерелий вычисляется по формуле:

$% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBa % aaleaacaWGRbaabeaakiaacIcacaWGUbGaaiykaiabg2da9maaqafa % baGaamytaiaacIcacaWGKbGaaiykaaWcbaGaamizaiaacYhacaWGUb % aabeqdcqGHris5aOGaeyypa0ZaaSaaaeaacaaIXaaabaGaamOBaaaa % daaeqbqaaiabeA8aQjaacIcacaWGKbGaaiykaiabgwSixlaadUgada % ahaaWcbeqaamaalaaabaGaamOBaaqaaiaadsgaaaaaaaqaaiaadsga % caGG8bGaamOBaaqab0GaeyyeIuoaaaa!5488! \[{N_k}(n) = \sum\limits_{d|n} {M(d)} = \frac{1}{n}\sum\limits_{d|n} {\varphi (d) \cdot {k^{\frac{n}{d}}}} \]$

Нужно доказать формулу произведения:

$% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci % GGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHEisPaeqaaOWa % aebCaeaacaWGobWaaSbaaSqaaiaadUgaaeqaaOGaaiikaiaad6gaaS % qaaiaad6gacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHpis1aOGaaiyk % aiabg2da9maalaaabaGaam4AamaaCaaaleqabaGaamOBaaaaaOqaai % aad6gacaGGHaaaaiaaigdacaGGOaGaaGymaiabgUcaRiaadIfacaGG % PaGaaiikaiaaigdacqGHRaWkcaWGybGaey4kaSIaamiwamaaCaaale % qabaGaaGOmaaaakiaacMcacaGGUaGaaiOlaiaac6cacaGGOaGaaGym % aiabgUcaRiaadIfacqGHRaWkcaWGybWaaWbaaSqabeaacaaIYaaaaO % Gaey4kaSIaaiOlaiaac6cacaGGUaGaey4kaSIaamiwamaaCaaaleqa % baGaamOBaiabgkHiTiaaigdaaaGccaGGPaGaaGzbVlaac6gacqGHsg % IRcqGHEisPaaa!6D7E! \[\mathop {\lim }\limits_{n \to \infty } \prod\limits_{n = 1}^n {{N_k}(n} ) = \frac{{{k^n}}}{{n!}}1(1 + X)(1 + X + {X^2})...(1 + X + {X^2} + ... + {X^{n - 1}})\quad n \to \infty \]$

С чего начать лучше?

[alcoholist]

левая часть от $n$ не зависит и от $X$ не зависит, а в правой есть и то и другое

[Deggial]

aspair, сформулируйте задачу корректно, иначе тема пойдёт в Карантин.

[aspair]

Формула взята отсюда Necklace_(combinatorics)

У нас известно:

$% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBa % aaleaacaWGRbaabeaakiaacIcacaWGUbGaaiykaiabg2da9maaqafa % baGaamytaiaacIcacaWGKbGaaiykaaWcbaGaamizaiaacYhacaWGUb % aabeqdcqGHris5aOGaeyypa0ZaaSaaaeaacaaIXaaabaGaamOBaaaa % daaeqbqaaiabeA8aQjaacIcacaWGKbGaaiykaiabgwSixlaadUgada % ahaaWcbeqaamaalaaabaGaamOBaaqaaiaadsgaaaaaaaqaaiaadsga % caGG8bGaamOBaaqab0GaeyyeIuoaaaa!5488! \[{N_k}(n) = \sum\limits_{d|n} {M(d)} = \frac{1}{n}\sum\limits_{d|n} {\varphi (d) \cdot {k^{\frac{n}{d}}}} \]$

Берем произведение:

$% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci % GGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHEisPaeqaaOWa % aebCaeaacaWGobWaaSbaaSqaaiaadUgaaeqaaOGaaiikaiaad6gaca % GGPaaaleaacaWGUbGaeyypa0JaaGymaaqaaiaad6gaa0Gaey4dIuna % kiabg2da9maaxababaGaciiBaiaacMgacaGGTbaaleaacaWGUbGaey % OKH4QaeyOhIukabeaakmaarahabaWaaSaaaeaacaaIXaaabaGaamOB % aaaadaaeqbqaaiabeA8aQjaacIcacaWGKbGaaiykaiabgwSixlaadU % gadaahaaWcbeqaamaalaaabaGaamOBaaqaaiaadsgaaaaaaaqaaiaa % dsgacaGG8bGaamOBaaqab0GaeyyeIuoaaSqaaiaad6gacqGH9aqpca % aIXaaabaGaamOBaaqdcqGHpis1aOGaeyypa0ZaaCbeaeaaciGGSbGa % aiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHEisPaeqaaOWaaSaaae % aacaaIXaaabaGaamOBaiaacgcaaaWaaebCaeaadaaeqbqaaiabeA8a % QjaacIcacaWGKbGaaiykaiabgwSixlaadUgadaahaaWcbeqaamaala % aabaGaamOBaaqaaiaadsgaaaaaaaqaaiaadsgacaGG8bGaamOBaaqa % b0GaeyyeIuoaaSqaaiaad6gacqGH9aqpcaaIXaaabaGaamOBaaqdcq % GHpis1aaaa!84DC! \[\mathop {\lim }\limits_{n \to \infty } \prod\limits_{n = 1}^n {{N_k}(n)} = \mathop {\lim }\limits_{n \to \infty } \prod\limits_{n = 1}^n {\frac{1}{n}\sum\limits_{d|n} {\varphi (d) \cdot {k^{\frac{n}{d}}}} } = \mathop {\lim }\limits_{n \to \infty } \frac{1}{{n!}}\prod\limits_{n = 1}^n {\sum\limits_{d|n} {\varphi (d) \cdot {k^{\frac{n}{d}}}} } \]$

Тут necklaces получили это:
$% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci % GGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHEisPaeqaaOWa % aebCaeaacaWGobWaaSbaaSqaaiaadUgaaeqaaOGaaiikaiaad6gaca % GGPaaaleaacaWGUbGaeyypa0JaaGymaaqaaiaad6gaa0Gaey4dIuna % kiabgIKi7oaalaaabaGaam4AamaaCaaaleqabaGaamOBaaaaaOqaai % aad6gacaGGHaaaaiaaigdacqGHflY1caGGOaGaaGymaiabgUcaRiaa % dUgadaahaaWcbeqaaiaaigdaaaGccaGGPaGaaiOlaiaac6cacaGGUa % GaaiikaiaaigdacqGHRaWkcaWGRbWaaWbaaSqabeaacaaIXaaaaOGa % ey4kaSIaaiOlaiaac6cacaGGUaGaey4kaSIaam4AamaaCaaaleqaba % GaamOBaiabgkHiTiaaigdaaaGccaGGPaGaaiilaiaaywW7caGGUbGa % eyOKH4QaeyOhIukaaa!6A09! \[\mathop {\lim }\limits_{n \to \infty } \prod\limits_{n = 1}^n {{N_k}(n)} \approx \frac{{{k^n}}}{{n!}}1 \cdot (1 + {k^1})...(1 + {k^1} + ... + {k^{n - 1}}),\quad n \to \infty \]$

Как это получили не пойму. Никак не пойму откуда $% MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AamaaCa % aaleqabaGaamOBaaaaaaa!3826! \[{{k^n}}\]$ вперед вышло.