¸ðÇÔ¼ö (Generating Functions)
º» °ÀÇ¿¡¼´Â ÁÖ¾îÁø ¼ö¿ (an) ÀÇ ¼ºÁúÀ»
±Ô¸íÇϱâ À§ÇÏ¿© µµ¿òÀÌ µÇ´Â »õ·Î¿î ÇÔ¼ö¸¦
Á¤ÀÇÇÏ°í, ±× ÇÔ¼öÀÇ ¼ºÁúÀ» ÀÌ¿ëÇÏ¿© ¼ö¿ (an)ÀÇ
¼ºÁúÀ» ¾ò´Â °úÁ¤À» ¾Ë¾Æº¸µµ·Ï ÇսôÙ.
Á¤ÀÇ 1. (an)ÀÌ ÁÖ¾îÁø ½Ç¼ö¿ÀÏ ¶§, ÇÔ¼ö

¸¦ ¼ö¿ (an)ÀÇ ¸ðÇÔ¼ö(generating function)¶ó°í
ºÎ¸¥´Ù.
|
¿¹¸¦ µé¸é,

À̹ǷÎ, ÇÔ¼ö

Àº ¼ö¿ { ( -1)n}ÀÇ ¸ðÇÔ¼öÀÌ´Ù.
¶Ç ¾Õ¿¡¼ ÇнÀÇÑ ÀϹÝÈµÈ ÀÌÇ×Á¤¸®¿¡ ÀÇÇÏ¿©

À̹ǷÎ,

Àº ¼ö¿ { 2n } ÀÇ ¸ðÇÔ¼öÀÌ´Ù.
ÀÌ ¶§, ÁÖ¾îÁø ±Þ¼öÀÇ ¼ö·ÅÀº Ưº°È÷ °í·ÁÇÏÁö
¾ÊÀ¸¸ç ´Ü¼øÇÑ ¹«ÇÑÇÕÀÇ ¸ð¾çÀ¸·Î¸¸ ÀÌÇØÇϵµ·Ï
¾à¼ÓÇÑ´Ù.
´ÙÀ½Àº ¿©·¯ °¡Áö ¸ðÇÔ¼ö¸¦ ³ª¿ÇÑ °ÍÀÌ´Ù.

ƯÈ÷ (3)½ÄÀº ´ÙÀ½°ú °°ÀÌ ±¸ÇÒ ¼ö ÀÖ´Ù.

½ÇÁ¦·Î (*)´Â ¹«Çѱ޼öÀÇ ¼ö·Å°ªÀ» ±¸ÇÒ ¶§ È°¿ëÇÒ ¼ö ÀÖ´Ù.
Áï,
Àº ºñÆÇÁ¤¹ý µîÀ¸·Î ¼ö·ÅÇÔÀ» º¸ÀÏ ¼ö ÀÖ´Ù. ±×·¸Áö¸¸ ±×
¼ö·Å°ªÀº ´Ù¸¥ ¹æ¹ýÀ¸·Î´Â ±¸Çϱ⠾î·Æ°í À§ÀÇ ½Ä (*)¿¡ x =
1/3
À» ¾çº¯¿¡ ´ëÀÔÇÔÀ¸·Î½á ±¸ÇÒ ¼ö ÀÖ´Ù. Áï,

¹®Á¦ 1.
(1) ¹«Çѱ޼ö ÀÇ °ªÀ» ±¸ÇϽÿÀ.
(2) ¹«Çѱ޼ö ÀÇ °ªÀ» ±¸ÇϽÿÀ.
|
ÀϹÝÀûÀ¸·Î ÇÔ¼ö f °¡ ¼ö¿ (an)ÀÇ ¸ðÇÔ¼ö
x = 0 ¿¡¼ ¹«Çѹø ¹ÌºÐ°¡´ÉÇÏ´Ù¸é, Maclaurin
±Þ¼öÀü°³¸¦ ÀÌ¿ëÇÏ¿© ¸ðµç ÀÚ¿¬¼ö n¿¡ ´ëÇÏ¿© °ü°è½Ä

ÀϹÝÀûÀ¸·Î ¸ðÇÔ¼ö´Â ÀÌ»ê¼öÇп¡¼ÀÇ ¿©·¯ °¡Áö
counting ¹®Á¦¿¡¼ ºó¹øÈ÷ »ç¿ëµÇ´Â µµ±¸ÀÌ´Ù.
½ÇÁ¦·Î ¾Õ °Á¿¡¼ ¹è¿î ºÐÇÒÀÇ ¼ö P(n)À̳ª
ÇǺ¸³ªÂî ¼ö¿ F(n) µîÀº ¸ðÇÔ¼ö ÀÌ·ÐÀ» ½á¼ ¼ºÁúÀ»
±Ô¸íÇÒ ¼ö ÀÖ´Ù.
ºÐÇÒÀÇ ¼ö P(n)Àº xÀÇ ¸è±Þ¼öÀÇ °öÀÇ °è¼ö·Î¼ ±¸ÇÒ
¼ö ÀÖ´Ù. Áï, Àü°³½Ä

¿¡¼

À̸ç ÀÌ´Â ºÐÇÒ
1 + 1 + 1 + 2 + 0 + 4 + 4 = 13
À» ÀǹÌÇÑ´Ù.
¶Ç À§ ½Ä¿¡¼ ¹«Çѱ޼öÀÇ °ö

¿¡¼ÀÇ xnÀÇ °è¼ö´Â P(n)ÀÌ´Ù.
(´Ü, P(0) = 1·Î ¾à¼ÓÇÑ´Ù. )
¹®Á¦ 2. ÀÚ¿¬¼ö nÀÇ ºÐÇÒ P(n)ÀÇ ¸ðÇÔ¼ö¸¦
±¸ÇϽÿÀ.
|
¿¹Á¦ 2. anÀ» nÀ» ¼·Î ´Ù¸¥ ÀÚ¿¬¼öÀÇ ÇÕÀ¸·Î
Ç¥½ÃÇÏ´Â ¹æ¹ýÀÇ ¼ö¶ó°í ÇÒ ¶§, ¼ö¿ (an) ÀÇ
¸ðÇÔ¼ö¸¦ ±¸ÇϽÿÀ.
Ç®ÀÌ. ¾Õ¿¡¼¿Í °°ÀÌ ´ÙÇ×½ÄÀÇ ¹«ÇÑ °öÀ¸·Î ÀÌÇØÇÒ
¼ö ÀÖ´Ù.
Áï, 5¸¦
2 + 3, 1 + 4, 5
¿Í °°ÀÌ Ç¥½ÃÇÒ ¼ö ÀÖ°í, µû¶ó¼ a5 = 3 À̹ǷÎ
ÀÌ´Â ¸ðÇÔ¼ö

¿¡¼ÀÇ x5ÀÇ °è¼ö·Î ÀÌÇØÇÒ ¼ö ÀÖ´Ù.
|
¹®Á¦ 3. ¼ö¿ (an)Àº 200, 300, 500¿øÂ¥¸® ¿ìÇ¥¸¦
ÇÕÇؼ n¡¿100¿ø ¸¸Å »ç´Â ¹æ¹ýÀÇ ¼ö¸¦ ³ªÅ¸³½´Ù.
ÀÌ ¶§ ¼ö¿
(an)ÀÇ ¸ðÇÔ¼ö¸¦ ±¸ÇϽÿÀ.
|
ÀÌÁ¦ Fibonacci¼ö¿ {F(n)} ÀÇ ¸ðÇÔ¼ö f (x) ¿¡ °üÇÏ¿©
¾Ë¾Æº¸µµ·Ï ÇսôÙ.

¾Õ¿¡¼ ¾ð±ÞÇÏ¿´µíÀÌ ¸ðÇÔ¼ö f (x)´Â ´ë¼öÀû, Çؼ®ÀûÀÎ
¹æ¹ýÀ» »ç¿ëÇÏ¿© ¿©·¯ ¹æ¹ýÀ¸·Î f (x)ÀÇ °è¼ö¸¦ ±¸ÇÒ ¼ö ÀÖ±â
¶§¹®¿¡ ¼ö¿ÀÇ ¼ºÁúÀ» ¾Ë ¼ö ÀÖ´Ù.

µû¶ó¼

À§ ½ÄÀ» °¢°¢ ÀÌÇ×Àü°³Çϸé

À§ÀÇ ½ÄÀº ¾Õ¿¡¼ ±¸ÇÑ Fibonacci ¼ö¿°ú ´Ù¸¥ ¹æ¹ýÀ¸·Î
Ç¥ÇöÇÑ ½ÄÀÓÀ» ¾Ë ¼ö ÀÖÀ¸¸ç, ¾ÕÀÇ °Í°ú °°Àº Ç×À» °¡ÁüÀ» ¾Ë
¼ö ÀÖ´Ù.
¹®Á¦ 4. ´ÙÀ½ ¼ö¿ÀÇ ¸ðÇÔ¼ö¸¦ ±¸ÇϽÿÀ.

|
¹®Á¦ 5. ´ÙÀ½ ÇÔ¼ö·Î »ý¼ºµÈ ¸è±Þ¼öÀÇ ÀϹÝÇ×ÀÇ
°è¼ö¸¦ ±¸ÇϽÿÀ.

|
|